Doug, it all depends on what you mean by "minor" repairs! If you don't want to perform welding (rust repair) and painting (big $$$), etc., then with your budget I'd suggest a Spridget, or a Spitfire or GT6, maybe a few others.
You can purchase a PERFECT Spridget (not a Bugeye) for $8-$11,000. In fact, a near-concours example! No muss, no fuss, get in and drive to the Brit car show and get a trophy. And, probably as much fun to drive as any LBC.
Spits/GT6s, same deal. Spitfires are a bit less and certainly no less fun to drive, a real deal for the dough, I think. GT6s in good condition are increasingly harder to find, especially MKIs and MKIIs. Great driving fun, if you don't need a drop top.
Also in your price range are a few others like Heralds (not so sporty, but fun!), maybe an MGBGT (only IF you're careful and lucky). Also, Sunbeam Alpines are in your price range as well. IMHO, a very lovely and often overlooked LBC. The body styling is really attractive and a real head-turner. There's a nice Alpine GT on eBay right now . . .
Some TVRs are in your ballpark, too. 2500Ms are going for around $8-12K for decent examples. A few nice looks 280s models in that range have been on eBay, too. Very cool cars.
And don't forget Minis: you can get a nice example at the top range of your $11K budget, but it may take a bit of shopping. Prices have risen on Minis recently, and continue to rise.
Most, if not all of the TR-6s I see in the under $11K range need work, and many have rust. TR-6s are no longer an entry-level LBC, but there are plenty out there and you might get lucky.
Again, it all depends on your personal definition of "minor". Very subjective.
It's definitely a buyer's market, and I think this trend will continue for the near future. Watch the key sales markets, grab someone who knows LBCs to give you a good second opinion on your pre-purchase inspection, and GET ONE!!!
Let us know . . .