The cloudiness is probably nothing to worry about, it is most likely a bacterial bloom and this can last several days to several weeks. It is not connected with cycling as such.
To quicken the cycling, adding bacteria is the only way to do it, other than with live plants. You can seed the bacteria from an established tank (your own, do not ever use media from a store or another aquarist unless you don't mind adding pathogens that could be dangerous). The bacterial supplements do much the same thing, and there are several on the market. Tetra's SafeStart is highly recommended. I have used Seachem's Stability with success. There may be others that work to some degree. Note that these do not instantly cycle, but they do speed up the colonization of the nitrifying bacteria.
Dr. Tim's One and Only is an instant cycling product that I have never used, but you add ammonia and this might not be advisable if you will need to be adding livestock. I cannot comment on how cycling may or may not affect axoltls as I have no experience with these.
The plant method involves some fast-growing plants, and floating plants are ideal for this. They are literal ammonia sinks, with the added benefit that nitrite is not a by-product so that is not a worry.