+1 for salvage shopping.
Find one that's been nailed on the side or back, with low mileage and engine in tact. Not only will you get a decent engine, but then some spare parts off of the donor car for anything else that comes up later.
Doesn't take much to write a car off these days, especially if the airbags have gone off. There's some good enough vehicles out there on salvage sites and/or auctions that aren't too bad off. 
Or just buy a similar replacement running MINI of the same generation as what you have now. Enjoy that one. But keep the one that you have and use that crapped-out one as a parts car for the running good one. 
Happy hunting

gera99vans is a SPAMMER

10 year old car? One option is to start shopping for its replacement. Other than that, I'd consider the new engine, especially if it has its own warranty.
 (Okay, that is an opinion, not advice! Hope it works out for you.)

Start shopping the salvage yards.

Not much help here - but you might try contacting one of the MINI clubs in WA to see if they can recommend a MINI friendly shop in your area. 

WASHINGTON

Inland Northwest Mini Club

Puget Sound MINI Motoring Club

Salish Sea MINI Motoring Club

Seattle Area Mini Owners Association

      I have a 2010 R56. My extended warranty just ran out last July. All the work that has been done to her was always done at a Mini Service Center (Seattle). Before my warranty expired I took it in for and extensive check up and they gave her an excellent bill of health. Fats forward to two weeks ago when she all of a sudden started losing power while driving on the freeway. She could barely do 30 on a slight incline. I managed to get her home eventually. I took to a local shop near me (closest Mini shop is 2 hours away) and they can only manage to hook it up to an OBD reader which spat out codes for random misfires on cylinders 2, 3 and 4. I called Seattle Mini service and they told me to bring her in, and I can still drive her if I feel comfortable. I made it about halfway before she really started having a hard time, and got her towed the rest of the way. That was last Friday. Yesterday my adviser called me to inform me that valves 3 and 4 are broken and when they a scope down 3 they found gouges and claims that it's not fixable and it will require an engine swap. I ask him is there anything Mini can do since my warranty just expired. Today he called me back and the only used engine he can find has 50k miles on it and only $500 cheaper than a new one. It's going to be around 6-7 grand without labor (Mini will eat the labor). My car's bluebook is less than that.  
      Does anyone know of a mechanic who knows how to rebuild a Mini engine for a reasonable price in or around Island County in Washington state? 
      Any advice?