My daughter was in an accident this summer; it was her fault; no one else was hurt. Our insurance paid the Medical payment coverage. Can I get more?

If it was her fault, and no one else was negligent or responsible, and if there were no manufacturing or similar defects that contributed to the accident or aggravated the damages, all she gets get is what her (or your) own policy provides.

As I have not read your policy I have idea what else your policy may provide, but if there is a $5k cap on medical damages (usually the policyholder had an opportunity to buy a higher limit), I sure doubt she can get more from her/your insurance company.