Be aware that Checkin is very rigorous about this and the carry-on weight and size strictly enforced. 10kg seems a lot but the size is limited to that of a large briefcase. As has been noted above a handbag is regarded as your 'piece of carry=-on luggage'.
If you are going to London for 3 nights I suggest you would really need to bring a check-in suitcase containing both your clothes, shoes, toileteries etc.? The prices quoted on the Ryanair website are of course for transporting the checked-in luggage both ways so its not excessive.