The landlord probably didn't know about the mice, he doesn't live there, so wouldn't have been around to see them. Only if he'd seen evidence (droppings and bits they've chewed) will he know, unless the previous tenant knew and told him.
We've had mice twice now.
2yrs ago we had a whole mouse family, put traps down (caught at least 10 mice), tried the sonic deterrent (they don't work, don't waste your money), filled in all the gaps where they could get in. Nothing worked.
Called our landlord, they called in a pest control company, who came within a couple of days, put down proper industrial poison and that killed them all humanely.
The other thing about this poison was that once the mouse picks it up, they carry it back to the nest and pass it on to the other mice, it also mumifies the mouse corpse so that it doesn't decompose and smell bad.
After that had no more mice until a month or two ago, when the weather started getting colder. Only seen one and heard scratching. Thanks to our previous filling of gaps, they're not getting to our food (they will chew food packets) but they're still in the house which is annoying.
Told the landlord again and this time we're having the council's pest control people come round, but because it's council we're having to wait for someone to come round.
My advice is make sure you don't leave any food or even bits of food about or on the floor (clean up after cooking and eating). This is what they come in for really, if there is no food, then their main reason for being there will go.
Put down traps, either the traditional snap shut trap (although this means you have a body to deal with) or the humane traps that will trap a live mouse (you then have to take them far away to set them free). Bait with peanut butter or chocolate spread. Don't bother with cheese it just smells bad.
Give your landlord a chance to sort it out, if they won't spring for a pest control company, either contact the council or offer to go halves on an exterminator.
Remind them it's in their interest because if they breed it will cause bigger problems; mice chew anything, cables, wood, plastic pipes, causing havoc to a house.
Good luck.
