I had the same problem on the later version ignition where that "Key in Ignition" switch is located on the front. Yes, you can solder it. You'll need to pop the retainer nuts off to get the switch of the main body. When you do, you'll destroy them and won't be able to re-use them. So you can try to find some locally, but I couldn't; even try to order some assorted ones online - no luck. So if you can't find any, then just get yourself some tie-wraps and tie it back on.
If you open the switch gently, it will come apart without any trouble. You'll want to clean the contact points of the switch while you have it apart.
If you're not worried about the "Key in Ignition" buzzer, then you don't even have to bother with it.