Might be worth seeing if any pest controllers locally are cheaper than your council - they sometimes are.
Having lived in a couple of old houses, one terrace house in a city and one surrounded by fields, both of which were incredibly attractive to mice, I count myself a bit of a mice expert.
I think you need both a short term and a long term plan and the advice you have had from PP is good. You must block up as many entry points as possible with wire wool. If you don't do this then you are largely wasting your time. The classic access points are broken air vents in brick work and the small gaps around where pipes come into the house, especially the water main. Also loft spaces. Do it now because when the weather turns colder in the Autumn, more mice will be looking for a warm spot to over-winter.
I'd try a combination of traps and poison. Have these out 365 days a year in inaccessible places like the loft and under kitchen cupboards even after you have stopped seeing mice. Check them regularly and you will soon see which areas are being used. Some mice are wary of new things so have them out all the time and in the same locations. Remember to push them against a wall, mice tend to run next to walls. I have never known mice move poison though they do tend to leave small stones (why I have no idea).
You don't need fancy bait for traps, I found mice would always take dry bread anchored with a dab of peanut butter (if you don't stick it down they get very good at pinching the bait whilst leaving the trap unsprung). Classic snap traps are the way to go. They are dirt cheap and mice are less suspicious of them than the fancy ones which require a mouse to go into a box. Wear gloves to set them, mice are wary of the smell of human hands.
If you can identify particular runs then I do think sticky traps are useful. I have caught mice which refuse to take bait this way. I used to put them down in the evening once the children and dogs were in bed and take them up in the morning. Any mice which were caught got a swift biff on the head - kinder than drowning.
My ultimate solution though was, I'm afraid, a cat, or two actually. Try and get one brought up on a farm or whose mum is a known mouser, apparently they learn from their mums. Our black and white moggie who was a kitten from a feral stray mother moved in and the mice moved out. I still have bait and traps down but never any sign of activity now. The only mice she catches now are in the garden and thankfully she never brings them in.