Removing Starter Solenoid from 1987 Accord
 

This is a project car I acquired this summer, it's still a work in progress but the most pressing issue right now is that I can't get it started. When I first acquired the car it started up fine, then the starter began slipping after cranking for a second or two. Now, when you turn the key the starter just spins freely.... it acts like the solenoid is seized or faulty (no click). I want to take it off and have a look, but I could only get as far as the starter motor. Once that was removed, I realized that to get at the solenoid I have to remove the entire starter bracket, and for some reason it's fastened with screws. Of course, they're seized and there's no way to apply enough force to loosen them as the screwdriver just slips off. Why on earth would Honda put screws in such a hard-to-reach location, and how on earth am I supposed to get them off? I can't even drill them out because there isn't room to get at them with a drill. Heck, I can't even reach the lower two with vise-grips!
I'm sure someone else has had this problem and I'm hoping they can tell me what to do.

You said when you turn the key the starter motor spins, but quickly? If that is the case, then the solenoid is working.

You may want to remove the distributor cap and turn the engine by hand using a 19mm socket on the crank bolt. See if the rotor turns when you do this.

Spinning quickly could also be because the starter drive is not "advancing" to the fly wheel. Could be the solenoid or the starter drive......but to be safe make sure the t-belt did not break.

edit: then, sorry I have to ask, I need to replace the starter on an 88 LXi auto soon, found a rebuilt unit for ~$60. Going to just get that, has a lifetime warranty.

When I apply power directly to the starter solenoid, there's a spark to indicate current flow but nothing else happens. Wouldn't that be a sign the solenoid is faulty?

Do you get the big mechanical CLUNK of the solenoid pulling in? It might be bad contacts within the solenoid (which are replacable). Look for some posts about that by HondaDude (he put some pictures of it too).