Welcome to the club!!!!!!!!!!!!! You got a bunch of options with this one and frankly this one isn't as bad as it looks. You will need to get a little clearance by unhooking the straps that hold the wiring harness in place. It looks like it could come out as easy as a good clamp with a vice grip. You can also buy these sockets that cut into and bite down like hell and help. It is going to take some good hand strength to do either but its not that bad.

P,S. This maybe because this is not the first time this bolt has been out. Fasteners have a coating when virgin and this prevents corrosion. Most mechanics do use any kind of coating when going back together. I use everything from #2 non hardening sealer, copper coat, anti seize, nuclear industry anti siege( for certain Ford engine plugs) at lastly just a couple drops of oil. In a pinch you can pull the dip stick on the engine or tranny and put a couple of drops. If you ever go back in you will appreciate it. 

Good Luck

You can drill a small hole in the center of the bolt. Then use a "left hand" drill bit, aka "easy out" to try to get it to turn. Take your time. 

Otherwise you can also go to an auto parts store, maybe even Walmart and buy a damaged bolt / nut extractor set. 

They are sockets that have reversed teeth to bite the head of the bolt to turn it out. Obviously you can't reuse the bolt.