Fran I could never get it to fully dissolve but it’s also never affected the results of the experiment so I wouldn’t worry about it too much. As long as enough of the potassium periodate dissolves you’ll be in business.
I did look it up and at room temperature the solubility is only 0.42 g per 100 mL of water but this does go up to 7.87 g per 100 mL at boiling so maybe heating the solution would help?