Washing machine connector leaked and water damage in adjoining bedroom (wooden floor, skirting boards, walls) - what to do?

Hi everyone, looking for some advice on water damage in a rented apartment. Moved to Germany a year ago and have a rented apartment which has been problematic (broken toilet, sound issues etc).

Yesterday I came back from a day out to find that there was water all over our bathroom floor and some in the adjoining bedroom. The water came from the connection of the washing machine inlet hose and the cold water tap (the washing machine was not in use at the time). I have a video of it dripping out. This has not happened before.

We have cleared the water and started drying the areas but there is minor water damage to a small area of floor (30x40cm), creeping damp on two walls next to the damp floor (roughly 20cm high but not equally spread) and to the wooden door and door frame to the bedroom. There is also a patch of creeping damp on the other side of the wall in another room (roughly 5x5cm). The floor and skirting boards are dry and the damp is disappearing following gentle heating and dehumidifier this afternoon.

The washing machine was professionally installed by the company we bought it from (Samsung) and we have had no issues since. I have liability insurance and contents insurance (this explicitly covers damage to floors and walls from water damage as well).

I have done my research and am aware I need to let my landlord know as soon as possible (i.e. tonight!) but I wanted to check what would likely happen when I do and what is the best thing to do. I have found conflicting answers online. Any help would be very appreciated! Our landlord is incredibly unreasonable and despite the apartment having many documented issues, acts like it is perfect (when it isn't). I am very afraid from what I have read that he will try and pin all he blame on me. I can't afford to pay any damages right now as my partner and I have just had our first child and money is tight. Will our insurances cover it? What do I say to the landlord? Thanks in advance!