What factors led to the Qing’s 100 Days’ Reform ending in a coup led by Cixi?

Right, I've held off on this long enough. What went wrong in 1898 and why?

Timelines

While it is tempting to start in 1898, we really need to account for the progress of the reform movement in the preceding decade in order to understand where it came from. To do that, I’ll need to provide a timeline of events or else we may be hopelessly confused when we step back and take a more thematic approach.

Aug.-Oct. 1884: French warships destroy the Fujian fleet in harbour as part of the Sino-French War. Dockworkers in Hong Kong refuse to conduct repairs on them, leading to a general strike and followed by a major riot.
Autumn 1888: Kang Youwei, a reformist Cantonese scholar who had been present in Hong Kong during the Sino-French War, petitions the Guangxu Emperor for legislative reform in response to Western aggression.
1891-4: The Guangxu Emperor studies foreign languages at the Tongwen Guan, and reads up on Western history and political science.
April-May 1895: Following the Treaty of Shimonoseki at the conclusion of the Sino-Japanese War, Kang and his protege, Liang Qichao, draft the 公車上書 Gongche Shangshu, a petition calling for the rescinding of the treaty and the continuation of war with Japan. 1300 students in Beijing sign the petition.
Summer 1895: Kang founds the 中外公報 Zhongwai gongbao ('Sino-Foreign Gazette'), a current-affairs bulletin distributed alongside the official government gazette.
August 1895: Kang founds the 強學會 Qiang xuehui ('Strength Study Society'), a group of thirty reformists including members of the gentry, the bureaucracy and some elements of the Western consular and missionary communities. Among them is Zhang Zhidong, Viceroy of Liangjiang, who officially sponsors its Shanghai branch.
February 1896: The Qiang xuehui is officially proscribed as an illegal political association and its newspapers are also banned. Zhang Zhidong rescinds his financial support for the Strength Study Society.
August 1896: Liang Qichao becomes chief editor of the 時務報 Shiwu bao ('Current Affairs Report') in Shanghai.
October 1897: The 國聞報 Guowen bao ('National News Report') is founded in Tianjin.
1896-8: Industrial development and educational reform in Hunan under Zhang Zhidong, now Viceroy of Huguang.
Autumn 1897: Liang Qichao and three other Cantonese scholars arrive in Hunan to assist in educational reforms. While there, Liang distributes anti-Manchu propaganda, largely pertaining to atrocities during the Manchu conquest of China such as the Yangzhou Massacre of 1645.
Winter 1897: The 南學會 Nan xuehui ('Southern Study Society') is founded in Hunan. Following the German takeover of Qingdao and the Jiaozhou Bay Concession in November, Liang Qichao petitions Governor Chen Baozhen to advise that Hunan should secede from the Qing, firstly to protect itself from foreign conquest and in turn to be the origin of a campaign of national liberation, in emulation of Satsuma and Choshu domains in late Tokugawa Japan. On recommendation from the Guangxu Emperor’s tutor Weng Tonghe, Kang Youwei is invited to Beijing.
24 January 1898: A meeting is arranged between Kang Youwei and reformist officials including Weng, Li Hongzhang and Ronglu at the Zongli Yamen. The bureaucratic reformists are dismayed by Kang’s radicalism.
Spring 1898: Hunanese conservatives and moderates begin publishing anti-radical materials, including Zhang Zhidong himself, who suspends official support for the reformist newspapers. Kang submits three memorials to the emperor, who also reads his older publications. Six new Study Societies are founded in Beijing, some associated with particular provinces (Guangdong, Sichuan, Fujian and Shaanxi) and some with particular intellectual approaches (知恥 Zhichi ‘Knowing Shame’ and 經濟 Jingji ‘Economics’).
12 April 1898: First meeting of the 保國會 Baoguohui (Preserve the Country Society), recruitment for which focusses heavily on candidates for the triennial metropolitan examinations, though it only ends up having 189 official members, and lacks the high-profile support of the Qiang xuehui.
April-May 1898: Much of the Baoguohui disbands after the exams, and the mobilisation of conservative and moderate opposition leads its officially being banned by the censors.
Summer 1898: A formal petition to Hunan Governor Chen Baozhen leads to the formal proscription of the radical reformists, culminating in the banning and destruction of Kang Youwei’s pamphlets in August.

So to begin with, it needs to be understood that the process of radical reform that occurred in Beijing in 1898 was neither a bolt from the blue nor the culmination of a continual upward trend of reformist sentiment. Rather, the reform movement had always taken the form of a series of fits and starts, with each wave of reformism seeing significant momentum before its radical characteristics became apparent, leading to it being proscribed by the authorities. However, many of these authorities were in fact erstwhile reform supporters, but supported more moderate forms of it. I’ll be referring back to pre-1898 events again later, but that gives us a good enough background of the broader situation to then go over the events of the Hundred Days’ Reform itself:

11 June: The Guangxu Emperor issues a proclamation signalling the court’s commitment to reform.
15 June: Weng Tonghe is dismissed, likely by Cixi’s faction.
16 June: Kang Youwei is summoned for an audience with the emperor. This ends up being the only meeting between the two.
July-August: Edicts pertaining to educational and economic reform are issued and promulgated.
3 July: The emperor announces the establishment of an Imperial University, Government Printing House and Translation Bureau.
18 August: The emperor orders the provinces to dispatch students to Japan for higher education.
Late August: Several offices in the metropolitan and provincial governments are abolished.
4 September: All six top officials in the Board of Rites are dismissed and replaced with a disproportionately radical board of seven men, three being Han Chinese associates of Kang Youwei or Liang Qichao and one of the four Manchus also having close ties to Kang. Four young radicals including Tan Sitong and Liu Guangdi are made secretaries to the Grand Council.
5 September: Radical reformist Zhang Yuanji proposes the abolition of Han-Manchu segregation and the effective dissolution of the Eight Banners.
11 September: Kang Youwei proposes the formation of a new imperial army under Yuan Shikai by expanding his nascent Beiyang Army.
12 September: Conservative official Zeng Lian petitions the court to execute Kang, Liang, Tan and Liu. Kang confides to Bi Yongnian, a friend of Liang Qichao’s, that he believes the emperor to be in danger from the dowager empress, and signals his intent to solicit support from Yuan Shikai in protecting him.
13 September: The emperor announces his intention to ‘open his palace for discussing changes of government structure'.
14 September: The emperor announces that the residence and professional restrictions on Bannermen are to be lifted.
15 September: The emperor sends a message to the reformists expressing his disdain at apparently being stifled by Cixi, who has just vetoed a plan to reopen the 懋勤殿 Maoqin dian, an audience hall used by early Qing emperors for holding emergency meetings.
16 September: Yuan Shikai receives formal promotion and instructions to expand the Beiyang Army.
18 September: An imperial censor, Yang Chongyi, sends a secret communique to Cixi accusing the radicals of associating with the Japanese and British and soliciting support from them. After reading the emperor’s message of 15 September, Tang Sitong attempts to solicit the assistance of Yuan Shikai in having Ronglu assassinated and Cixi placed under house arrest.
19 September: Cixi arrives in Beijing and sits in on the Guangxu Emperor’s subsequent audiences.
20 September: Yuan Shikai leaves Beijing for Tianjin, and while there possibly reveals to the Manchu nobleman Ronglu the details of the anti-Cixi plot.
21 September: Cixi launches her coup. Soldiers under Ronglu storm the Forbidden City. The emperor is placed under house arrest.
21-24 September: Six key radical reformers are arrested including Tan Sitong and Liu Guangdi. Kang Youwei and Liang Qichao escape.
26 September: Cixi nullifies all the major changes since 11 June.
28 September: The arrested reformers are executed without trial.

So that’s our timeline of events, bringing us back to the original question: How do we get from a reformist wave in June to a military coup in September? While there’s no single factor, I’ll outline here the most salient perspectives by which to view the causes of the movement’s collapse.