Probably best to go back in the morning with it. My doctor's surgery is two bus rides away and it takes about an hour to get there. You're lucky yours is so close.
That's what he said too - he was with that doctor for years - yet you just fill in a piece of paper at the doctor near you - bob's your uncle and you are that doctor's patient.
If they have not provided a box or slot to put things in (maybe they've had bad experiences with local louts posting things) then clearly you have no choice but to return with your letter when they open. It's possible they collect their usual mail from some redirected PO postbox ?