My 96 triple suffered the same fate but got to the point where it would not start at all. The starter would simply spin and nothing.
I got the bike some years ago and its done me well (28,000 miles for me) so I figured it had a good run. I let the bike sit for about a year until deciding to give it a nice makeover.
I shopped around, and as others stated parts run you around $350 to do everything yourself (coolant, gaskets, oil, etc.). All in all I've spent about $1,100 on the bike, but I must say it is a blast and would do it over again in a heartbeat.
I started by getting a nice jack and rigged up a system to hold the bike from a beam in my garage. Dropped the engine, cleaned and repainted everything external while it was off, and put it all back together. Fixing the starter should run you about ~10 hours (I'm no mechanic, but it wasn't too bad). If you take it to a shop, figure labor off of that (8-10 hours).
While I had the engine out, I also cleaned, rebuilt the carbs, and fitted up new air filters and re-jetted. No matter what, I highly recommend you do the same as well as the gains are amazing. The induction sound on acceleration makes riding SO much more enjoyable, and maintenance is a breeze as you do not have to remove the carbs to simply clean the filter. Now I just loosen a clamp and voila. I also gutted the mufflers as this is the hooligan hot rod of the bike family for me.
I logged in simply to give you hope. This was my first motorcycle so there is some sentimental value to it, but it is also one of my favorites. I would highly recommend taking a shot at it yourself. While vague, the service manual led me correctly. There are also a few assemblies to choose from, I believe one of them is a new revision and is a bit beefier. Since my local dealership has always been very honest and more than fair to me (3 triumphs from them, and the speed triple goes there for inspections & tire mounting) they were more than helpful on the financial side as I purchased a bunch of goodies from them to refresh the various items that needed attention (new turn signals $10 for 4, 6 piston calipers and pads, new controls (off the next generation though) with heated grips.
All in all, I myself spent around 50 hours working on it and I have loads of videos and photos of things along the way to help me with assembly. While it seemingly took forever, I must say it was one of the best decisions I’ve made in my life both the pride and enjoyment of tackling such a project really sheds a new light on the bike. Here’s hoping your tachometer doesn’t quit on ya next!
If you would like any guidance, don't hesitate to ask