This is the only logic board IFIXIT has right now: [product|IF108-022-2] There is also a 1.8 GHz board but you’ll need to look online as they are very fleeting as many people want them. Here’s the IFIXIT guide to put it in: [guide|6313]
I wouldn’t spend to much on this system as before you know it it will cost more than a new system. As far as a new system… Rumor has it Apple will be introducing a newer Air system either at WWDC next week or later in the fall. If your system is working right now I would hold off to see what they do.