Hi there,
As Julia (Bellabelloo) suggested, it would be best to move Rexx into a container (an icecream tub, tupperware container, bucket, etc) of shallow de-chlorinated water (enough water to cover Rexx's back), and put this tub in a cool place in the house (the lower to the ground, the better, as lower ground is cooler than higher ground). Keeping Rexx in a small tub of water will enable you to keep tabs on him, and monitor his feeding. Is there anywhere cool, dark, and quiet in the house that you can keep Rexx's container?
While being kept in the tub, Rex will need a full water change each day with fresh de-chlorinated water. When changing his water out, just scoop him up in a net, empty the soiled water out, pour fresh dechlorinated water into the now empty tub, and quickly (but gently) put Rexx back into the tub. He'll be a little put out by this, but should recover, if left well alone after the water change, put in a nice dark and cool place, and left with a nice juicy worm to enjoy in the privacy of his tub.
.