"I don't have a mouse" just shouldn't be part of the conversation. Unless you have material reason as to why you can't use one (having the computer on your lap in the plane / train, being discreet in a school ampitheatre), it should be an issue only during the 25 minutes it took before you bought a new one.

With that said, I'll go with your restriction and say that your computer is weak enough that you switch your expectations and treat it as an emulation machine.

Even on PS1, many games don't use the right stick at all and only use the left stick as a D-pad, so you should be able to play keyboard-only even for a fair number of action games. You can also just not bother with straight-up action and play long-winded turn-based RPGs instead, like the Final Fantasy or Pokémon series.

As a bonus, many of the things that could make these games boring or frustrating will be lessened by the qualify-of-life features of emulation (quicksaves and fastforwarding).

Your computer would make a great PS1 / GBA / SNES machine, and other systems too but let's focus on these for the best compromise between low-tech and "modern" features. If so, then just head over to YouTube and look for videos like "top ten RPGs on the PS1" or whatever your wish is, and see if there's anything you like.