In some respects, it is incorrect to portray this as the Qing 'enlisting' Europeans, as this would imply their seeking out Europeans to fight for them. In practical terms, it was mostly Europeans insisting on employment with the Qing, and the Qing generally not refusing them. Rarely were the Qing taking the initial initiative in creating such European-led forces, but by and large they would then work with these forces after they were established.
Over the course of the later years of the Taiping War, several European-led forces offered their services to the Qing, and we ought to distinguish between a few types of foreign military support: elements of regular military forces engaged against the Taiping, fully-foreign volunteer units, and so-called 'Disciplined Chinese' forces with foreign officers and Chinese troops. The dividing lines between these could, of course, be rather fuzzy: for instance, a foreign volunteer unit might reorganise into a 'Disciplined Chinese' force, and many of these forces were officered by members of the patron powers' regular armed forces without resigning their commissions. But as a general schema, this division should serve us well enough. For my part I will focus primarily on 'Disciplined Chinese' forces as well as their antecedents in the foreign volunteer contingents.
We also ought to be careful of treating the Qing as a monolithic entity. At the very least, we ought to consider three different layers of authority: the local level, where officials held immediate oversight; the provincial level, where governors, viceroys and their staff were responsible for the broader operational conduct of the war; and the imperial level, where the court held control, however nominal, over the overall conduct of foreign relations. A small volunteer contingent might, if it came on the radar at all, operate at the discretion of a middling functionary like a city mayor, whereas 'Disciplined Chinese' forces would generally be directly subordinated to provincial-level authorities; the imperial court rarely intervened meaningfully, but might still be drawn into situations where there may have been a particularly acrimonious dispute over some issue relating to a foreign force. There is a world of difference between the Shanghai mayor turning a blind eye to the European merchants organising small uniformed militias, and the Viceroy of Liangjiang or of Min-Zhe co-opting a brigade of European-led Chinese troops to support their provincial armies.
Some aspects of the foreign intervention in the Taiping War have been well studied, the Ever-Victorious Army especially; others less so. Rather than give a comprehensive overview, I will instead focus on a few of the better-studied cases, and specifically on the side of the 'disciplined Chinese' contingents, as also covering direct foreign intervention would make this post monstrously long.
The Ever-Victorious Army, or EVA for short, is a particularly interesting case in that it essentially went through three distinct organisational phases, each with a somewhat different relationship to Qing authority.
In June 1860, two Americans, Frederick Townsend Ward and Henry Burgevine, petitioned Shanghai officials and elites, including the banker and city treasurer Yang Fang, to permit and fund the creation of a force called the Shanghai Foreign Arms Corps, initially consisting of around 100 American, British, and other European volunteers equipped with modern rifles, later reorganising into a mostly Filipino force with European officers numbering around 250. While it saw limited success in capturing the town of Songjiang in mid-July, it was essentially disbanded after suffering a serious defeat at Qingpu near the end of the month, and Ward, severely wounded in the engagement, would end up convalescing overseas – his exact whereabouts are unknown – for the better part of a year. In September, the force was dissolved by the Shanghai circuit intendant.
The statement that Ward and Burgevine petitioned for the EVA's establishment does not necessarily communicate the full story as such. While it is true that the proposal came from them, Shanghai's circuit intendant (essentially equivalent to the city mayor) Wu Xu had a particular interest in getting as much military force in the area as possible, as the British and French had, so far, not made any kind of unequivocal declaration that they would defend the Chinese part of Shanghai against a Taiping attack. In the desperate middle months of 1860, when no major Qing armies stood between Nanjing and Shanghai and a Taiping capture of the latter seemed imminent, any help, from whatever origin, was eagerly accepted. Critically, it was eagerly accepted by Wu Xu, but not by his superior the prefect of Suzhou, who refused to forward his petitions to the court. The Foreign-Arms Corps would be established at Wu Xu's discretion, and maintained thanks to Yang Fang's personal networks in the Shanghai banking world, rather than the official sanction of the Qing court.
After Ward returned to Shanghai in March 1861, he attempted to assemble a new force on similar lines as the first, now named the Shanghai Foreign Legion, which suffered yet another defeat at Qingpu. British and American authorities in the International Settlement, concerned that the activities of the adventurers constituted a breach of foreign guarantees of neutrality in the conflict, arrested Ward and disbanded the Legion, and he and Burgevine went into hiding until late in the year. When they re-emerged, they proposed a new organisational structure for the third attempt, wherein they and the other foreign adventurers would instead serve as officers for a force otherwise composed of Chinese troops (save for a company of Filipinos as Ward's personal guard). This force, initially named the 'Foreign Arms Corps' again, achieved its first success at Songjiang in February 1862, and soon after it would be rechristened the 'Ever-Victorious Army' by Ward's Shanghai financiers.
At this stage in the proceedings, the Qing government as a whole was more receptive to foreign help than it had been. Peace had been reached with Britain and France (whereas the two countries were still at war with the Qing back in 1860), and the military defeats leading to that peace deal had been a potent demonstration of the power of modern weapons in trained hands. There were also lucky breaks in that there were significant political changes at the provincial and imperial levels. Li Hongzhang, a protege of the Qing's main field commander Zeng Guofan, was appointed governor of Jiangsu in early 1862, and took over control of the war in the eastern theatre thereafter; Li, like Zeng, appreciated the benefits to be gained from adopting foreign weapons, and had a knack for diplomacy as well. More importantly, the fiercely anti-foreign Xianfeng Emperor had died in August 1861, and the similarly anti-foreign council of regents for his son was overthrown 3 months later by a considerably more pro-foreign group consisting of reformist elements in the imperial house, principally the late emperor's brother, Prince Gong. So both international and domestic politics had shifted to where foreign mercenary assistance was broadly desirable.
But this was helped along in no small part by the reworked structure of the EVA. The Qing were wary of permitting too much in the way of direct foreign intervention for fear of having to grant further concessions in supplementary arrangements, and were also unwilling to allow recaptured cities to be garrisoned by foreign troops, an issue which had previously complicated the working relationship between Ward and the Qing during the Foreign Arms Corps days. The EVA's officers being private citizens volunteering on their own initiative mitigated the former issue because they did not act officially on behalf of a government, while their troops being Chinese dealt with the latter because even an EVA garrison would be majority-Chinese, assuming they would supply a garrison at all as opposed to various second-line contingents like the Green Standards or local militias. It also helped to some extent that Ward and Burgevine had evaded arrest by renouncing their US citizenship and petitioning to be recognised as Qing subjects, which gave Qing authorities somewhat more power over the force than it might have done otherwise.
An added, and not entirely unintended benefit was that Ward had achieved somewhat of a rapprochement with the foreign authorities in Shanghai, thanks to the general opinion of the international community decidedly souring on the Taiping. While the British and French were wary of supporting the Qing directly, they were willing to lend their services in theoretical support of the EVA, and so EVA and foreign regular troops operated in tandem in what was sometimes termed the 'thirty-mile radius' campaigns, so named because officially, British support for Ward would not extend more than 30 miles from Shanghai. Having Ward around thus gave Qing authorities a somewhat roundabout way of gaining additional military support from European regular forces.