I think it's generally not advised to do work experience in your own surgery and esp not with your own doctor, so perhaps look at contacting other nearby surgeries instead?
I think phoning up the surgery or going through HR would be the correct way of asking about work experience, GPs are on a tight time schedule and you appt is really only to discuss your health
I've heard many people find it a struggle to get work experience in the NHS as contact isn't always frequent or thorough but I advise to just keep on persevering, it took many of my friends a while but they've got there eventually