"Very hard to get an appointment "

I have called the surgery yesterday morning to make an appointment to see GP as my daughter has high fever for the last 3 days, been vomiting, also she has been complaining about her ear pain. The nurse over the phone told me it is only viral and there is no need to come but I insist I wan to see a doctor and she booked an appointment! (Anywhere in leaflets you can read if person don't get better within 3 days you should go and see a doctor) A doctor has diagnosed my daughter with "TONSILLITIS" and prescribed her antibiotics for 10 days! I told the nurse how she can diagnose it is only viral over the phone without seeing my daughter?? Why I am not allowed to have an appointment to see a GP when I am paying for NHS??? Why is it so difficult only in this "surgery" to get any appointment at all?? I am so upset as my daughter is very ill, can hardly walk to toilet and would have to suffer who knows how many more days if I didn't insist to see a doctor. This is very upsetting as this has happened few times already and I am registered in this surgery only last 6 months...