It won't work cause Shimano jammed 10 cogs in the same space 8 used to go, so the cogs are more closely spaced on 10spd.
But that also means you can swap your 8spd cassette for a 10spd one, and it'll work then, with a 10spd chain. Derailleur won't need changing. You don't need to use Ultegra either, cause all the groups are compatible, pretty much.
Tell you what though, I bet you can get your old 8spd levers working again by flushing them with WD40. STIs are notorious for retiring early when they get old, cause the factory grease gums up and stops some tiny pawls from engaging. Factory grease aside, they should work for a long time.
I read that people got 8-speed shimano setups running very well with campi 10 speed brifters. In fact, I was going to try this myself on a new bike build a while ago but ended up going all-campi and using the shimano parts on another project. See the link for a good article on shifting mixes.
You could also try a set of used 8-speed shifters on e-bay. I picked up some 8-speed 105 brifters that work well. On ebay, I noticed someone was selling 8-speed indexed downtube shifters. That'd be the cheapest option. But, if you really want the upgrade to 10 speed, you could also try 105 (instead of ultegra). 105 is good. My wife runs it on here bike, and functionally, once tuned, I don't really notice a difference between it and the ultegra on mine. All of it runs pretty well; keeping your drivetrain cleaned and lubed will make a biggest difference in functioning. You should only need to replace the shifters/levers and your cassette. Of course, after spending big money on the shifters, you may decide to go ahead and replace some other parts too...
Cheapest 10 speed STI option in the US is the Nashbar branded microshift units. They put them on sale all the time. Still need to change other parts to go to 10 speed. You can also get your 8 speed stuff rebuild pretty cheaply. There is a guy on facebook that rebuilds any of the 7 and 8 speed stuff (no Sora).
Why not just try it, and if you can't get it to work, then get a 10spd FD.
If it doesn't want to work, I bet some judicious cage-tweaking could sort it, as long as the actuation ratio's close enough.
But are you prepared for more finnicky indexing thanks to the closer spacing? If Campy 10spd shifters will work your Shimano derailleur over 8spd nicely, I reckon that's definitely the way to go. Peeps reckon those 10spd chains wear out fast, too...