As a follow-up question: How did the British army model compare to the French and Prussian army models, and why did Chile not consider emulating the British model? How have the differences between the British, French, and Prussian army models affected the modern British, French, and German armies?
The French influence of the Chilean army predates 1858. In fact, the Chilean army was not built upon ideas of mid-19th century French thinkers, but actually on Napoleonic military organization and thinking together with 18th century Spanish military manuals.
The origins of the French influence can be seen during the Chilean War for Independence (1813-1823). French officers and soldiers who had once served Napoleon and who emigrated to South America after the battle of Waterloo found a new way to continue their military careers in the Ejercito de los Andes. The experience and knowledge that these men brought with them was disseminated within the army and helped to bring French influence to the nascent Chilean army. That the Chilean army was organized after the French model was formally established in the 1830s. In practice, this meant for example that the Chilean army organized artillery separate from infantry and cavalry, that the higher Chilean military leadership was decentralized in peace but centralized in war, and a focus on the French column formation in line tactics. In addition to these practical considerations, Chilean soldiers were dressed in French-style military uniforms. This Chilean army of the 1830s, formed after the French model, was affirmed as it triumphed over the Peruvian-Bolivian confederation in the War of the Confederation (1836-1839). The Chilean army was here to stay.
So, what happened?
Leading up to the War of the Pacific (1879-1884), the Chilean army remained stagnant in its ways. With the exception of acquiring modern weaponry, there was no move away from military organization or tactics that had been at the height of their popularity in the early 19th century. To fight in compact Napoleonic formations with modern repeating arms in 1879 might have seemed like suicide, but to many Chilean officers, this was the way wars were fought and won. Some Chilean officers did instruct their men in dispersed tactics, drawn from contemporary conflicts in Europe, but these were few. Upon hearing a suggestion to train their men in dispersed tactics, Diego Dublé Almeida recalls hearing the commander of the Navales battalion, Martianio Urriola, saying that: "[T]he Chilean soldier didn't need the dispersed order to win at Chacabuco and Maipú, Yungay and Calama: it was enough to be a Chilean soldier to win and they won." Tactics that had been good enough in 1818 (battle of Maipú) and 1839 (battle of Yungay) were more than enough for 1879, argued Urriola and his sentiments were widely shared amongst Chilean commanders. As this contemporary illustration of the Battle of Tacna in 1880 shows the type of unimaginative tactics by Chilean commanders that were used during the conflict.
The debate surrounding the use of compact versus dispersed formations intensified during the war. As previously noted, some took the initiative on their own to train their men in it, but it was far from universal in the Chilean army. Despite these antiquated tactics, Chile again came out triumphantly over their northern neighbors in what has gone down in Chilean history as the most important conflict in their history. The last military manual drawing on French ideas was released in 1884. With victory secured and a terrible defeat forced upon their enemies, there were still questions about the Chilean military performance in the conflict and the future threats that might arise as a result of the war. With the potential that Peru or any other South American nation might want to attack Chile, president Domingo Santa María chose to initiate a military reform starting in 1885. The nation who would serve as a model for the Chilean forces fell upon the now unified German Empire whose victories in three separate wars (The Dano-Prussian War, Austro-Prussian War, and the Franco-Prussian War) stood out to Chilean observers.
The Prussian influence on the Chilean army reached its height between 1891 (after the Chilean Civil War) and 1906. It was during this period that Prussian officers like Emil Körner became Inspector General of the Chilean Army. Körner and his fellow Prussian officers, including a Swedish officer by the name of Wilhelm Ekdahl (who wrote a multi-volume work about the history of the War of the Pacific, completely in Spanish), set out to completely modernize the Chilean Armed Forces, carrying through a major overhaul in organization, weaponry, uniforms, tactics, strategy, and much more. Since this became the basis of the modern Chilean Army of the 20th century, we still see the remnants of the Prussian influence today in parade and dress uniforms, and even in the uniforms of the Chilean national police force (!).
Chile never fought another international war again. Yet it was the performance of the Chilean army in the War of the Pacific and the geopolitical consequences of the war that ultimately became the vital factor in modernizing the Chilean army for the 20th century. Now that Chile had one of the most successful and intimidating armies in South America, they had to be ready to take on any new threat that might arise as a result of their terrific victory. For this uncertain future, Chile turned to what what they believed to be the most powerful army in the world. Whether or not Chile would have performed better in another international war was something the world fortunately never found out.
Sources:
For an in-depth study of the late 19th century military reform in Chile, see The Grand Illusion: The Prussianization of the Chilean Army by Holger H. Herwig and William F. Sater. For the French influence, see Los soldados de Napoleón en la Independencia de Chile. 1817- 1830 by Fernando Berguño Hurtado and "El ejército de Chile en vísperas de la guerra del Pacífico. Una aproximación a su influencia francesa (1866-1879)" by Valentina Verbal Stockmeyer in Revista Universitaria de Historia Militar, Vol. 3, No. 5, 2014.