So I have a Nishiki Olympic 12 road bike that I keep at my mom's house in Italy. Since I live in New York, I only get to ride this bike once or twice a year when I go visit her. It is currently set up as a 6 or 7 speed freewheel, I think (this is the problem with trying to upgrade a bike that you only get to see on rare occasions) with Dura-Ace downtube shifters (used on friction setting). Hubs are Shimano 600 Ultegra HB-6207, rear derailleur is a Campy 990 (rare-ish 80s model) and front derailleur is a Campy 980. The crank is a Shimano 600 double, not sure about the chainrings. Here's a picture of the current set up:

So, I have a beat up pair of Shimano 105 ST-5500 integrated shifters (9-speed I believe), that were given to me and have been sitting around in my parts bin for years, that I would like to refurbish and install on this bike when I go to Italy in mid-December. The shifters need some work, small parts and new hoods, but assuming that they are usable, what else will need to be done to upgrade this bike to 9-speed, STI shifting? Will I need to replace the derailleurs with Shimano 9-speed compatible? What about the rear hub? Will I need a new wheel with a 9-speed freehub and cassette, or is there some way I could get the shifters to work with what I have on it now?

I do have a set of adjustable shifter cable stops, so at least there's that!

Thanks in advance for any and all advice/suggestions!