Hi, and welcome to TP. You haven't given much detail, and obviously a lot depends on your circumstances. Here are some thoughts.
Firstly, who has advised you to become a deputy? It may not be necessary. If your mum has a low income and savings, and doesn't own the house, then you might only need to apply to become a DWP Appointee in order to manage her State pension, Attendance Allowance, and any other state benefits.
If your mum has larger assets, then it is probably necessary to become an deputy if she is unable to grant Power of Attorney. That might still be possible. The donor has to understand in the moment what they are granting. They don't have to remember all the details afterwards. If you think she might still be able to grant LPA then check out the website of Solicitors for the Elderly to find a suitable solicitor who could act as professional certificate provider. You can do LPA yourself using the online form, but if there is doubt about your mum's capacity it's best to use a solicitor.
With regard to you both living in the same house, what do you think might be the issues that could create a conflict of interest?
Your mum needs to live somewhere, so she would pay for bed, board and utilities wherever she lives. You as Attorney or Deputy would need to work out what was a fair share for her to pay for household bills and upkeep of the property. Perhaps also a share of maintaining and running a car if she benefits from that.
If the property is hers and you don't pay rent then this might need to change. It all depends on your specific circumstances and who has paid for things over the years.
An issue that sometimes arises is where the parent might need to go into residential care but the son or daughter fears being made homeless, either because the home is rented or because it needs to be sold to pay for the parent's care. In that situation there is certainly a conflict of interest, but it isn't impossible to resolve it. It wouldn't matter who was the deputy, a decision would need to be made about how the parent's care was paid for. There could be a number of options.
I have rattled on excessively, and may be barking up the wrong tree.
I hope I have given you food for thought.
