Patters,
You asked for a "LED strip light that's multi coloured and can cover 10m" and that can wire into a "light ceiling rose".
I am presuming that your issue is that the device you are looking at has a 240V to 12V converter that has a plug on it (a wall wort), and hence this will not go into your light ceiling rose.
So my
first suggestion is that you get a 240V to 12V converter that is inline, i.e. there is a plug, a length of cable, the 240V to 12V converter, a length of cable, strip of LED lights.
Plug this into one of your wall sockets, get a length of extension wire for the 12V side, and route it up the wall to the strip of LED lights you have glued to the ceiling. As long as this length of extension wire for the 12V side is not too long then any voltage drop will not effect the brightness of the LEDs.
So my
second suggestion is that you get a 240V to 12V converter that is inline, i.e. there is a plug, a length of cable, the 240V to 12V converter, a length of cable, strip of LED lights.
Cut off the 240V plug, so you have a length of wire on the 240V side, and wire this into the ceiling rose. You will then have to make an (unsightly) clamp or box on the ceiling to hold the 240V to 12V converter. Wire the 12V side of the 240V to 12V converter to the strip of LED lights you have glued to the ceiling.
As you want changing lights to make a pattern, get a remote control for the strip of LED lights.
Not a recommendation, simply the first set that came up, but I would suggest something like this:
https://www.amazon.co.uk/dp/B071F6Z...1501507529&sr=8-1&keywords=led+strip+coloured
or this
https://www.amazon.co.uk/Non-Waterp...1501507529&sr=8-6&keywords=led+strip+coloured
from this page
https://www.amazon.co.uk/s/ref=nb_sb_noss/259-5939143-7087362?url=search-alias=aps&field-keywords=led+strip+coloured
And regarding your LED strip, extension leads and connectors (IF you get a RGB version) are here:
https://www.ebay.co.uk/sch/i.html?_....TRS0&_nkw=RGB+led+lights+connectors&_sacat=0
SFK