The cassette meachanisms of this age are all similar in how they work.
Pressing a button down, causes a metal slide to get puched to one side. The specially-shaped holes in this metal slide engage the pressed-down button(s) and lock them in place until the "stop" button is pressed (actually it resets the slider bar, so it releases the buttons).
What has most likely happened is that the hole that engages the button that won't stay down has become worn. You will have to check it. See how the latching mecanism works on a button that will stay down. You will see how the non-latching button SHOULD be working then.
If you are really lucky, a slight bending of the metal will get it working again. You might even be able to use an epoxy adhesive (like "Araldite") to fasten a new bit of metal in to replace any that is too worn.
If you are unlucky, the whole slider may need replacing and it might even be easier to replace the latching mechanism itself. You could possibly get a "doner" mecanism from a really old cassette player. The trick would be to get one where the buttons are spaced the same distance apart as the ones on the CKP-100.
Hope this might help a little?! I noticed today that service manuals are still available to buy, if you search online for them. Incase one might help you visualise how the mechanics work.
They are beautiful machines and well worth keeping going if you can! I don't think any modern mini-TVs have come close to the quality of this old-faithful from 1980!
TK_M, February 2008