Because Kings Cross station is one of the handful of big stations both owned and managed by the infrastructure operator Network Rail, the best source of information isn't National Rail Enquires (as normally would be the case), but Network Rail's own station page for Kings Cross, and more specifically the Kings Cross station facilities page
This contains quite a bit more showering details:
By the entrance to platforms 9-11.
Monday to Sunday 07:00 - 21:00
£5, pay at Excess Baggage, soap included. Towels available - deposit required.
This is almost all correct....
On arrival at Kings Cross, head towards platforms 9-11. Go past platform 9 3/4 (really!), and by the stairs up to the Parcel Yard pub is the Excess Baggage Company office. Go in there, pay them a fiver, and they'll give you a receipt.
Next, follow the corridor by the Excess Baggage Company signposted as Toilets. At that barrier, ring the bell, and show them your receipt. They'll get you a towel (no charge/deposit) and some shower gel, then unlock a hidden door in the appropriate toilet area (male/female). You then go in and lock the door.
The shower, at least on the male side, is in good condition and quite large. The non-shower area in the room is a little small, but does have 3 sturdy coat hooks. One downside - no shower curtain, so you'll need to hang most things up to avoid it getting wet!
Plenty of water pressure (maybe too much given no shower curtain...), nice and warm water, decent shower gel and pretty good towel. For a fiver, and after a long journey, it's not bad.