So you have turned the lights off on this skink because you cant play with him? Dont you think this will make its health worse? How are you maintaining the 95-100f baskinng spot BTS need?
Theres no point in medicating if you have already given up on basic husbandry.
As far as your question, the medication should say how to apply it. Meds taken oraly or externally can not be injected and vice versa. If it is infact a RI then you will need injections of 'something'. What the medication is, is usually dictated by a culture test done by a vet. This is all regurged information however, as I have no first hand experience with having to give reptiles meds.
I wish your skink luck. I wish for some things for yourself as well. Not as nice as the skink though.