Derailleur compatibility with Shimano 5500 shifters
You have many options for a replacement. Essentially any pre-11-speed Shimano road derailleur with a GS cage length will work, excluding Tiagra 4700 and pre-indexing/early indexing ones.

The rear derailleur actuation ratio is the same across many generations of Shimano groups. The earliest Dura Ace indexing groups used one ratio, now a footnote in history, then everything else used the same ratio (known unofficially as 1.7 in some sources) for 6, 7, 8, and 9 speed groups. Road 10 also used 1.7. The first break from this compatibility came with Mountain 10, and since then every speed generation has been its own thing, with the exception that Tiagra 4700 is a ten speed road group that uses the 11-speed actuation ratio. (There will probably be more like this going forward.)

For the most part, the ten speed era derailleurs that use the same actuation as your 9-speed-era Ultegra will drop in and work with zero issues. This includes any GS derailleurs from 5600, 5700, 6600, 6700, 7800, and 7900. You need GS because you have a triple. You could also use one from the 9-speed groups, like RD-5501-GS, etc. (That's the one that "matches" your shifters.)

It's an old group now but I believe Shimano is still producing RD-5701-GS as a legacy option for a high quality replacement for the compatibility you need. That matches the quality of the rest of the bike so it probably makes some sense to just get that one. RD-R3000-GS would be a functional budget choice. After what the bike has been through, make sure the hanger is aligned. It's also prudent most of the time to start out with at least a new cable and ideally new cable and housing with a new RD.

Using a less-than-9-speed derailleur with the correct actuation can work, but should usually be avoided because the precision might not be there for it to perform well.