Hello,
I think there is a luggage deposit/ locker point at Euston train station, though it's run by a company and can cost about £8 (I looked into it myself at one point). The British Library near King's Cross has lockers that would fit a backpack for £2 (?), though I feel bad just using their lockers and not visiting/ contributing.
As well as BL, the Wellcome Collection is just across the road from Euston and it has some fascinating medical science-based exhibitions. They also have a free cloakroom and cafe (though it is quite expensive e.g. over £5 for soup). Euston Road stop is nearby if that helps with changing lines.
P.S. It can be so difficult getting to Gatwick (I am from the Midlands) - will you post when you've returned to let other users know how you found the experience/ offer any tips, please?