Auxiliary verb reduction in not only - but also structure

This is a straightforward case of coordination at the level of complement to the auxiliary verb. The subject I, auxiliary verb did, and modifier not only apply to all three elements in the coordination. The elements in the coordination are plain infinitivals acting as complements to did.

Not only did I

[finish my homework, exercise, play some video games],

but I also cleaned the house.

Examples of similar constructions are readily found.

Nor did either Wiener or Humphrey's Executor suggest that Congress could not only create the independent agency , specify its duties, and control the grounds for removal of its members but could also itself appoint or remove them without the participation of the Executive Branch of the Government. (Supreme Court Opinion, BUCKLEY v. VALEO, 424 U.S. 1)

The issue on which we all seem to agree is that self-interest and the common good are two sides of the same coin, and that to survive as a sustainable planet we must not only create wealth , develop skill and talent, but ensure that no one is left behind.(International Development White Paper, Hansard, 3 may 2001)

When he preached, Thomas became wholly a man; not only did he laugh , crack jokes, sing and shimmy on the pulpit, but his severe stammer left him completely. ( Order and disorder on Smith Island; Sheehan, Kevin)

Not only did he write , produce and play on some of the most popular and influential songs of the disco era for Chic, Sister Sledge and Diana Ross alongside his fellow New Yorker, Bernard Edwards, but he has also created hit records for an eclectic array of artists, including Madonna, David Bowie, Deborah Harry, Duran Duran, Mick Jagger, Al Jarreau, Jeff Beck, INXS and the B-52s. ( CLASSIC TRACKS: Chic 'Le Freak' | - Sound On Sound)

As is clear from these examples, all verbs in the coordination have the same inflection.