Had to hardwire mine.
I don't think you'll find an AP that, when installed in your house (anywhere), will be able to propagate a usable signal into your shop under the conditions you laid out. If you are strictly opposed to running a cable, then a point-to-point wireless solution is about your...
Hard to get a perfect measurement by myself with a tape, but I'd call it about 80"...maybe a touch less.
It fits in/under our 7' garage door if that helps.