- My mother insisted that I should stay at home for a couple of days due to my illness.
This sentence is okay.
- My mother insisted that I am staying at home for a couple of days due to my illness.
This sentence is okay grammaticaly, but the meaning is off kilter: it does not say that your mother wanted you to stay at home.
What it means is that your mother was telling to someone (maybe even to you) that you are at home for a couple of days due to your illness. She was telling this in an insistent manner, but she wasn't commanding anybody to stay at home. She was trying to convince, not to command.
We can transform sentence 2 in a sentence equal in meaning to sentence 1:
My mother insisted that I stay at home for a couple of days due to my illness.
Such construction is called "mandative subjunctive" and is used to express a circumstance that is demanded, desired, etc.
The Wikipedia article on subjunctive points out the possible confusion between the indicative and subjunctive mood in sentences with insist, suggest and propose:
I insist that he is here (indicative, a forceful assertion of the fact that he is here)
I insist that he be here (subjunctive, a demand that the condition of his being here be fulfilled)