OP is renting tigger
OP, if you bleach it will come back. HG mould spray makes a real difference as it kills all the spores. I recommend it to every tenant I see, round here ALL houses have terrible condensation issues.
As others have said, wash all fabrics now, very hot wash, bleach include if possible. Clean down all surfaces, make sure you get the back of cupboards, underneath too if you can get the kickboards off (don't scream if it is all black under there, it does just wipe off!)
Get the LL back to look at the wall, window and gutters, stuff that is the fabric of the house, that will cost him a fortune in the long run. Point that out to him and the agent, slowly and carefully, whilst telling them that you have washed down the affected areas with specialist mould spray, to contain it.
You might need to learn a new way of heating and ventilating your new home. I've lived in single glazed, old, often run down houses for most of my life, I now own one! So I am used to the illogic of heating a house just to let the hot air out to ventilate. But you will eventually find which windows and doors you need to open to get a through draft that will solve much of the problem.
If you add that Karcher vac to the dehumidifier you'll be fully armed for the Condensation Wars. Once you have your routine you might love the house as much as the DCs and DH. My DH does the emptying of the dehumidifier and scraping of the windows whilst he makes our morning coffee, it takes about 5 minutes in all. Get yours up to speed with that and you should be much more on top of it.
As others have said, I love my quirky house and the dreaded black spot mould, which was rampant when we moved in, is no longer evident other than pale grey spots in places we haven't yet painted!