The problem is you are way overstocked for that sized tank. the mandarin in a 13.5 gallon tank needs the extra feeding so i get why you are doing 3-5 times a day but that is just overload on such a small tank. 2 clownfish and 1 springeri might even be too much in that sized tank.
Honestly best move i think would be to just buy a 20 gallon or larger tank and move everything over. Skimmer would definitely help for now though and can be moved to a larger tank after.
Here are some options for skimmers
https://www.marinedepot.com/aquamaxx-nf-1-nano-protein-skimmer-in-sump
Compact Filtration for Aquariums Ranging from 10-30 Gallons Footprint: 2.75" x 3.2" (3.5 x 4in w/Mount)| Reaction Chamber: 2" | Height: 13"Skimmer Pump: DCair 200
www.coralvue.com
No room for a quality protein skimmer? Think again. The Eshopps Nano Skimmer has a small footprint of just 4 1/2" x 2 1/4" x 11 1/2" high that allows convenient placement in small spaces- perfect all-in-one nano aquarium systems. The easy-to-install in-sump skimmer is powered by the...
www.marinedepot.com
Fiji Cube Nano Protein Skimmer is an excellent choice of nano skimmer that offers great skimming capability. The skimmer has a compact footprint which is designed to fit most all-in-one aquariums. The skimmer comes with height-adjustable mount that allows high flexibility in different water...
www.fijicube.com