There is no need to cool your water down to 5-10C. As long as you can maintain the temperature below 20C especially during the summer as the warmer temperatures can stress an axie. And sorry, Layla not sure where you get the idea not to feed for 10-14 days (Axies may not eat if the temperature is lower than 10C, ie if they are in the fridge for recuperation/because they're sick).
Katie, apart from checking temperature another thing that should be checked if floating occurs is testing your water parameters for ammonia, nitrite and nitrate. If levels of ammonia or nitrite are above 0 then a 20-30% waterchange should be done daily or every 2nd day to bring the levels down (no need for adding chemicals (apart from one that removes chlorine and chloramine from tapwater).
Also, one should never do a complete water or large waterchange as this can cause your tank to start cycling again, which can invariably effect your axie if you do not maintain regular (usually daily) partial 20-30% waterchanges depending on size of tank.
What substrate do you use? If your axie continues to float and you can keep his tankwater temperature below 18-20C that may help.
Otherwise if he continues to float or looks wobbly (leaning to side) placing him in a container of dechlorinated water, low enough for him to rest feet on bottom and not float, and placing him in the fridge should work. Place a lid on and wrap in a dishcloth (to stop temperature fluctuations and light flashes from normal fridge useage). Place a bottle of dechlorinated water in the fridge which you will use for complete daily waterchanges on his container. 5C is the minimum temperature for fridging an axie. Fridging an axie lowers its metabolism and will calm an axie down. You may find after a few days in the fridge that he comes right.