Walk the tire in to a auto tire shop, not the front desk, the back where the bays are. Ask whomever you encounter if they can put a plug patch in to repair it, and that you'll pay cash on the spot, no receipt needed, no concern about liability, You just want the tire fixed and it's on you after that.
I've done that and $20 later I had a well repaired tire with the patch plug in place. The guy scuffed up the inside with a die grinder, reamed the hole, coated the inside area with cement goo, inserted the patch plug, pulled the tail thru, smoothed it out on the inside, then coated the whole inside area with more cement goo for good measure. Never had a problem with that to the end of the tire, 6-8k miles. Yours may require a little reaming to get the patch plug thru, or they might opt to just put a patch over it. Go with what ever their experience suggests, this is nothing new on car tires. I'm sure they have seen this before.
edit - It's a small, strait hole. I personally would just install a sticky string and be done with it.