The cycling process can take from 3-6 weeks.
For your tank to be cycled, a beneficial bacteria needs to be present to break down the Ammonia and Nitrites into Nitrates. Ammonia and Nitrites are lethal to Axolotl and are produced from your Axolotls waste and decomposing food. Nitrates are a slightly less toxic by product. During cycling the beneficial bacteria will colonize your tanks surfaces such as substrate, ornaments, filter, and begin to feed on the Ammonia turning it into Nitrites, then bacteria will break down the Nitrites into Nitrates. The bacteria is known as a 'biofilter', as it filters the ammonia and nitrites into less harmful nitrates.
Once this cycle is established by the beneficial bacteria colonizing the tank, the Ammonia, and Nitrites will always be filtered by the bacteria into Nitrates, all you have to do is monitor you cycle by testing your water weekly so that when your Nitrates build up to around 40ppm you can perform a 30% water change to lower the Nitrates.
You seem to be stuck at the stage of your Nitrites not being filtered into the final NitrAtes. You will just need to be patient, and the cycle will continue. If you get excessive levels of ammonia and Nitrites do a 30% water change and then test your water again. You just want to be looking for a rise in NitrAtes then you know your Nitrites are begining to be filtered.
Make sure you are following the instructions that came with your testing kit perfectly, this way you will get true results.
Are you using dechlorinator?
Can you post images of your Axolotl in its separate container?
Axolotls will usually only develop fungus if there is a breach in their slim layer from damaged to the skin/gills, if the water quality is poor, and if the temperature is too high.