Ultegra 6510 shifter broken - upgrade to 10 speed?
Shimano groupsets get better over time so the latest R3000 Sora shifters are probably not as bad as you think. Certainly better than worn out Ultegra 6500 series.

You could upgrade to Tiagra 4700 10 speed, which again is probably actually better than the older Ultegra 6500. I test rode a couple bike with it and it seemed to comparable to the mix of Ultegra 6700 and 105 5700 I was running at the time.

I thought that 11 speed required a wider freehub body to accommodate the cassette, but I've read on this site that that is not always the case, so you might be able to go 11 speed.

About mixing and matching components: 

The difference between 9, 10 and 11 speed cranks is the spacing of the rings. Higher speed chains have a narrower outside width (inside with is the same) so the rings are slightly closer together. Opinion on this site seems to be that you can run a 10 speed rear and chain on a 9 speed crank without too many problems.

You can in fact run 10 speed rear shifters with a 10 speed cassette and 9 speed derailleur because Shimano 8, 9 and 10 speed road derailleurs do all use the same actuation ratio (except Tiagra 10 speed which used the 11 speed actuation ratio), so it you can find some used 10 speed Ultegra or 105 shifters in good shape those will work.

Unfortunately my knowledge stops at front derailleur compatibility, I've yet to figure out the actuation ratios for those.