Tricky one @Ben. I would be asking the landlord if they could improve the drainage. What kind of soil is it? Heavy clay?
One trick you might want to try is to aerate it with a garden fork to at least help the rain penetrate more.
Obviously the ideal would be to improve the drainage, but I wonder whether just adding some sand might help?
Thanks for all the suggestions Workshoppers! After trying a few "low effort" options, I finally followed @Johnnynl's advice... but called in the professionals to assist.
I've added a project diary and step by step photos in this post:
Hopefully somebody else can benefit from my experience!