The sentence is ambiguous, however, West Point was founded in 1802.
Cadets would not have founded West Point, so the sentence already switched to the context of what those cadets did (in 1828).
Academy officials determined the capsule was placed in the base of the Kosciuszko monument by cadets in 1828, 26 years after the academy’s founding.
I would have said:
> [...] base of the monument by cadets in 1828, 26 years after the academy's founding.