I find it incredible that people struggle with reading information and grasping what it clearly says.
OP's 40th post: "I don't work at the moment. I've not set a notice period. Shall I get that in writing to him?"
OP's 42nd post: "I've been pretty naive and didn't want to come across as rule driven. It's backfired and lesson learnt. He pays monthly."
In the absence of a formal notice period, the guidance from reliable sources states clearly that it is reasonable to align a lodger's notice period with the frequency with which they pay rent (in this case, monthly) as the Shelter link you posted says:
"What is reasonable notice?
Reasonable notice must be fair to both you and the landlord.
1 month is reasonable in lots of situations.
Sometimes it could be more. For example, if you have lived there a long time.
But it could be less. For example, if you and the landlord find it hard to live together or you have broken your agreement."
and from the .gov link I posted:
"Reasonable notice usually means the length of the rental payment period. For example, if rent is paid monthly, you should give one month’s notice."
There is nothing in what OP has posted that suggests the reasonable notice period she should give him should be anything other than a month - no longer and no shorter.
NOW you can stop going around in circles!