If you've got an analog cable signal, all you really need is an analog cable tuner. Your Mitsu almost certainly has one, and you could potentially run the cable TV to the antenna input and run audio output (stereo analog most likely, although if your set has an ATSC tuner for over-the-air HD you may even have a digital audio output) to the 990. Even with the speaker turned off on the TV, the audio output should still work.

Another option would be a spare VCR used as an analog cable tuner (cable TV in, stereo analog audio and composite video out to the 990), but the video signal path will probably be cleaner using the Mitsu's tuner.