I think my (very large) house plant has spider mites. I cut some badly affected parts off, but there are probably still some I can't see. Is there something I can do to get rid of all of them?

I agree with rachel.

An organic type approach is take a 32oz spray bottle add 2-3
drops of dish-washing liquid and several drops of hot sauce,the hotter the better, add water & shake. Spray all house plants to keep them from leaving one to go to the other.
Spray under leaves and on soil as well. Will not harm you or your plants. All soft bodied insects(white flies, aphids,spider mites) can't tolerate this mix and will die. Works on all plants in & out.