Here is the situation:

1. Cavalry approach.

2. Infantry form square.

3. Cavalry innitiates melee (can't "charge" as infantry is in square so low rate of success) and fails.

4. Infantry (stays in square so as not to risk getting caught in line) fires at cavalry (at low effectiveness because in square formation).

5. Repeat steps 3 and 4 over and over again with no resolution.

Other than artillery which is not around to alter things, am I missing something?

"Combined attack with infantry"!

That is historically correct. Almost always, attacks by cavalry against squares failed, but without support by their own cavalry or their artillery, the infantry would not be able to drive off the cavalry.