Hi Brian
As a full time carer, then you do not have to look for work.
To get the LCWRA element in your joint UC claim then your wife will have to apply for the WCA, in a joint claim you cannot have two LCWRA's awarded, I would advise you to seek help from your local Welfare Rights Organisation, they will be able to take all your circumstances into consideration;
advicelocal.uk .
Gary