Most the advise above is probably good, but you might consider sending it to Sullivan for a diagnosis...OR just buy one at a swap meet for $10. Might want to look for a Hobbico, actually.
Bought my Sullivan starter for $10 at a club swap meet and eventually bought the Finer Edge Mfg. adapter ($50) to use Makita's latest 18v. LiIon battery packs. It turned out that my Sullivan starter is an early production one (which means probably 30-40 years old). It worked fine on a 7ah 12v gel cell as sold by Tower and most hardware stores and hobby shops. But the cord was a hazard, and I really wanted an integral battery/starter per my Kavan starter (with optional Ni-Cad pack). Since Kavan stuff is now difficult to get (Fritz & Franz retired), I wanted a replacement ready to go.
I already had to put a new switch on the Sullivan, because the original got to where it wouldn't shut off, which is very bad. Good news is that parts are available. The bad news was that the adapter's instructions didn't quite work (because of its vintage), but I managed to get it together. It's a friggin' BEAST on 18v., and spins my OS .46 VF (AAC via Randy Aero) like it's nothing...and it's something!
The guy that makes the Makita adapters recommends a Hobbico, because it's an easier conversion. In case you might go that way. Otherwise, about the same. He developed it for R/C Helicopters, of course. The heli guys use a long extension shaft on their starter, so the Sullivan starter's switch staying in the "on" position is well known and extra bad news.
For full disclosure, I didn't have a Makita drill kit, but wanted one, and when I found out I could also use the batteries to run my starter, I was all aboard with that. I have used the heck out of my Makita drill this summer building fences and gates, and it's awesome! So's the Sullivan starter. But I might be looking around at the swap meets for a newer version. There's at least 3 of these Sullivan/Makita conversions in the NW Skyraiders.
Another FYI, you can buy aftermarket cups for various starters, some made from polyurethane, some from silicone, etc. Can also get ones shaped specifically for "needle nose" spinners, which I figure should give more "traction". Check Tower and figure out what you want and then check at your LHS.

Steve