You can't make a claim directly against someone else's insurer. You have to make a claim against them and leave it to them as to whether they wish to claim on their insurance.
Send a formal written demand for compensation to your landlord, including all of the following:
Your name and address
His name and address. (If he trades as a limited company, the demand needs to be addressed to 'The Company Secretary', not to your landlord by name, and sent to the registered address of the company, not to his home address or other business address).
The date of the letter.
The reason that the demand is being made.
The amount demanded.
A warning that, if the compensation isn't received within a specified time (I suggest 28 days) you will initiate court action for compensation plus costs.
Send the letter by recorded delivery, retaining a copy of the letter and proof of posting. Go online to check that it's been received, noting the date, time and signature.
If you don't get your compensation, use the online small claim's procedure:
https://www.moneyclaim.gov.uk/web/mcol/welcome
Chris