before everything else you need to be aware that your axolotl should be tubbed until the tank is dealt with.
I can assume that your axolotl has been in the tank since November so the tank should have cycled, if you are getting traces of ammonia or nitrites then either the filtration isn't large enough or there is waste in the tank/filter at the level that your nitrites are I'm inclined to say it is filtration issue.
the nitrates though have to be removed via water changes, plants or special nitrate removing filter media, it is always best to do a water change weekly, dechlorinating the water before adding to the tank. the percentage of water removed is the percentage of nitrates removed, test the nitrates before a water change to find out how much water needs removing.
looking at his gills he doesn't look bothered about the high levels although they do need to be dealt with asap.
using holtfreters in the water should offer some protection against high levels, info here
Axolotls - Requirements & Water Conditions in Captivity including move information on water parameters.