Are the LA paying the full cost of the CH or are they agreeing to pay up to their 'limit' and the difference is to be paid out of your mum's assets or by you? If this is the case I assume you would pay the 'top up' to the CH direct as it makes the LA admin easier for them - although not being in this position (yet) with MIL's care home I may be wrong.
The LA never pay the full cost of a care home. The state pension of a person will always have to be paid towards the fees as will any occupational pension if it exists. This is a misconception a lot of people have. In addition a certain amount will be paid from the capital between the upper threshold of £23,250 and the lower of £14,250 at a defined ratio.
In addition any top-ups cannot be paid from the assets of the person concerned as, if they could afford to do so, the LA would not have to be involved with payments. These top-ups have to be found elsewhere.
However, the answer to Roboandkate's query is as Wirralson has said, that, in most cases I believe you will pay the portion you have to pay from your mother's account, direct to the LA. I certainly do. The invoices come direct from the LA and I simply ring up and pay it.
in cases where the assets of the person being cared for (your mother in this instance) fall between the upper and lower threshold, being recovered from the assets of the person concerned.
However, it makes no difference whether you are talking about the upper or lower threshold, the payments will still be required, as I have already said, even once your mother's capital has fallen below the lower threshold.