Hi,
My computer was recently repaired, and worked fine... then stopped... then worked again perfectly for a month... then stopped again. I have figured out, that essentially the computer works fine, the battery is fine, and the AC adapter, and where it plugs into is fine. However, for some reason, the computer stops reading that an AC is plugged in, doesn't get power, and therefore, the computer dies, because I can't ever get the battery charged back up, and it's not accepting power from the adapter. What can be done? It is working now, but because it is not reading power, in about five minutes it will die again... and then for no rhyme or reason that I can figure out-- it won't read that there is an AC adapter... and then w/ out me even touching it, a couple of days, or hours, all of the sudden it will read that there is an adapter again, and will work normally again. Is there anything I can do to resolve this in this moments when it does work?

The indicator on the Compaq is a "charing" indicator. It really doesn't indicate that something is plugged in.

First recalibrate your battery. Put laptop in safe mode with the battery in and the adapter out and wait until it shuts off by itself.

Then fully charge.

Then try to determine how long the battery lasts by itself.
You might get 1 to 1.5 hrs on a good battery.

Try flexing in a controlled way the power cable and see if the indicator lights. A voltmeter is a better test.
The HP adapter usually fails at the ends of the low voltage cord. Either at the brick or shortly before the plug that plugs into the laptop.

My current re-build of the power brick includes a super flexible cable on the DC side. I connected that to a short piece of an iGo power cable (I had those fail too. They have almost no flex) and an iGo Tip. Of course, it's not an iGo supply for anything, just the laptop.

I tried using Radio-Shack's TIPS, but the quality and wire gage won't support the Laptop's power requirements.

I've had bad lifetime experiences with the Battery packs.

Ok sound to me like you have a bad charger or your DC JACK that is disengaging from the motherboard; DC jack is where the AC adapter gets plugged into.
Best thing to do is jiggle the adapter end while it's plugged in and watch the battery charging indicator on the laptop to see if it goes off and on.
If that is not your problem then first thing to do it change the adapter and see if issue goes away. You can purchase and inexpensive adapter from eBay. From $10 to $20 just enter model number on eBay search engine with the word charger next to it.