With a an arrival not until 1330 and a departure at 2115, that would mean a day room would only be used from, say, 1500 until 1800, so the value is questionable.
Planning to visit one of the tourist attractions with a fractious jetlagged child is not, IMHO, particularly advisable.
I would just stay flexible and find a quiet corner of the terminal where Jnr could crash out for a couple of hours, and then maybe get out if the terminal for an hour or so and get some fresh air and take a tube trip to Acton Town and back. If they haven't done it before, just riding the tube is usually excellent entertainment for five year old boys.