There's no convenient Underground station. You need to get to Cricklewood (National Rail) station.
St Pancras station is next door to Kings Cross station, so you can start from there. There are 4 trains per hour, direct to Cricklewood, with a journey time of just 11 minutes. The return fare is £3.10 but you can also use a Oystercard (where you'll have just £2 deducted from the available credit) or a Travelcard.
On leaving Cricklewood Station, turn left (onto Cricklewood Lane). The first turning on your left is Claremont Road, with the care home just a short distance along that road, on the right-hand-side. (It's only about 2 minutes walk from the station).
Chris