How to repair this damaged air compressor

(A picture of the broken piece would help.)

Option 1. (By far the best answer) Buy a new hacksaw. You won't be able to fix this without removing the threaded piece that still remains in the hole, then fabricating a new piece to hold the blade. This will involve adding threads to the new piece. If you don't have the tools to do this work, then you will need to obtain them, costing you more than a new hacksaw frame.

Option 2. Remove the threaded part that remains inside the hole. Usually this involves drilling a small hole in it, then you must buy a tool for removing screws that have been broken off in a hole. (Its usually called an ez-out, or something like that. A hardware store will have them.) You will insert that tool in the hole you have drilled. It will jam in place with a coarse reverse thread with sharp edges, and allow you to back out the threaded piece. It may help to drip some wd-40 in there as you are trying to back out the threaded piece.

Drilling that hole may itself be problematic, since there is a reasonable chance that part of the hacksaw was hard steel - by design to withstand the loads it must take to hold the blade. Worse, you must drill a hole into a surface that is jagged.

You may also be able to get the threaded piece out by other means, but you must do all of this without damaging the female threads in there, otherwise you will NEVER be able to fix the new piece in there.

As I said, unless you have a few important tools, this is all something that may be more easily repaired by simply buying a new hacksaw frame. Money solves everything, although a hacksaw frame is not that expensive. And if you don't know how to get that screw fragment out, then you probably don't have those tools, nor will you use them often enough in the future to merit buying them. At the same time, it is good to be able to repair things like this, so if you have the interest and the will to do this sort of work enough in the future, then go for it. But don't be at all surprised if you find it takes you considerably more time and expense than it is really worth to simply buy that new hacksaw frame.

I would strongly encourage you to replace this unit (the Siemens component, not the entire AC) and replace the ends to those electrical wires. Simply put if the rust situation is that terrible I wouldn't be at all surprised if the integrity of the unit is already compromised, i.e. ready to fail you at any time. Your yellow wire is almost certainly why it's not working now, but it's a pain to fix it, and then have to fix it again only a few days/weeks later.

This is not a normal amount of rust; a great deal of humidity is getting to it, more than usual. I'm a Florida resident who had a 12 year old AC unit at one point that had been through a number of hurricanes and tropical storms, and it was only slightly oxidized, no rust. (Though I had to regularly open it for service because for whatever reason ants loved to commit suicide by throwing their bodies in between the electrical contacts)

To fix the rust you'll need to find where it's getting exposed to so much humidity from. Likely a seal or gasket is bad. You can either replace the broken seal/gasket or there are after-market products used as essentially spray on sealants. I recommend fixing it properly and replacing whatever seal or gasket is broken, but in a pinch squirting some sealant into the location that's leaking (with the AC OFF and given enough time to dry) will resolve minor leaks (depending on the severity).