Vaguely similar questions have been asked, but I believe this situation is unique enough that I’m looking for some specific advice. I have a floor that is framed with 2x8s, 16 O.C., spanning roughly 11 feet between a central built-up beam and the outer sill plate. The joists “rest” on a standard (code minimum, blech) 2x2 ledger strip. The hardwood floors are a bit bouncy and it bugs me, so I plan to stiffen the floors by adding some sisters, or (more likely) gluing and screwing a bottom flange to the joists (to avoid removing tons of mechanicals and solid blocking).
However, before I do that I want to address the fact that the joists seem very shoddily connected to the beam. In addition to the ledger strip, about half of them are toe-nailed (see first pic, ok, that seems fine), but the other half are not (see second pic). This strikes me as very sketchy. There are a few end nails into each joist from the built up beam, and on some of the joists these seem to be carrying the whole load.
I thought about pounding some joist hangers in and then prying off the ledger strips, but my thinking is that the fact that half the joists are already toe-nailed to the beam means that adding joist hangers to these might cause splitting and I don’t want to raise inspection red flags by having a weird mix-and-match setup of hangers and ledger strip. Some of the non-toe-nailed joists have small gaps between joist and beam, and some aren’t even fully resting on the ledger (see pic 3, !). I do believe this was probably sloppy cuts rather than separation after the fact.
How can I improve these joist-beam connections? Should I just add some 45 degree toenails and call it good? The joists are fundamentally in good shape and if possible I'm looking for something DIY-able.
Edit: As I suspected, I was able to confirm that there are 3-4 end nails driven into the joists that are not toe-nailed at least (confirmed by feeling in the gap between joist and beam with a shim). Not sure if this changes anything.