I've had experience with caring for small sick and injured skink when I was caring for and rehabilitating a rescued (very small) juvenile water skink (Lucky) and I was required to give him liquid food only as one of his injuries from the cat attack I rescued him from was a broken lower jaw and puncture wound to the corner of his mouth - so he had too sore a mouth to hand "hard" insects like crickets and mealworms.
I was giving him oral doses of antibiotics by syringe with a 24Gauge flexicatheter tube (no needle) and liquid calcium (CalciVet) the same way , and handfeeding him the guts of gutloaded crickets and mealworm lavae and mealworm pupae (the last two being his favs) ... he was initially about 5g and regrew his tail and thrived ending up about 20g.
Towards the end I was giving him as part of his diet a mix of VetaFarm lizard and Repcal Juvenile Beardie Pellets made up as a past and using a 14Guage feeding needle on a plastic syringe who's hole was drilled to give a 3mm bore.... this "worked" but was hard work with the syringe I had.
You need a syringe with at least a 2.5mm bore to get thick slurries to flow.
see
viewtopic.php?f=45&t=232687 , two methods of attack with syringe feeding
1) gently push the end of the feeding needle between lizard's lips and past the teeth just far enough in to be on his tongue (from the side of his mouth so you are not injecting straight into his throat) , inject small amounts of slurry into the mouth, this is a two person job, one to hold and restrain the lizard who will likely object.
2) use the feeding needle to inject the food directly into the lizard's stomache. see above link for how this is done.
The appropriate volume to give per feed, and appropriate size of the crop/feeding needle is given in the link.
I had success with mix of VetaFarm Lizard and RepCal Juvenile Beardie Pellets (1:1 mix) in enough water to cover them in small 25ml tub, soak, and them mix make an paste , I then added a little water at a time to make a wet slurry , this I gave by feeding syringe and feeding needle into the mouth a little at a time.
You could add Repashi BugPie made as slurry as well ,
and if you can source it , VetaFarm Critta Care for Reptiles
http://www.vetnpetdirect.com.au/vetafarm_critta_care_reptile
for a very sick lizard == Give approximately 5mL per 100g body weight, per feed.
Use a feeding by syringe for lizards and amphibians.
or Wombaroo Reptile Suppliment
http://www.wombaroo.com.au/reptiles/reptile-supplement
Mix made as a slurry 10g / 15ml water).