You can fly into Vasteras airport from the UK - it used to be Stansted, not sure if it still is. It's not too far from Eskilstuna, they only have one flight in and out in the morning and one later in the day as it's a really small airport, nice place though. It's one of the budget airlines that goes there - Ryanair I think.
You can rent a car at the airport but it is expensive, when I went car hire was £100 per day.
There are a number of hotels nearby, we stayed at the Ibis. Although the rooms were tiny it did the job.