Jancw hello and a warm welcome to Talking Point,
You are obviously doing wonderfully looking after your mother but everyone needs a break to look forward to. I'm sorry that there is friction between your siblings but working around it is obviously important for you to keep some life of your own.
For us respite was paid for partly by Social Services and we paid a contribution, this was calculated after an assessment by Social Services of needs and income. Basically if you aren't in charge of your mother's finances then speaking to your brother who sounds to be in charge of her money would be a good start, if he doesn't want SS involved as they would want to know exactly what is coming in and going out then it's up to him to pay for respite out of your mother's money. If he doesn't agree to that then you can contact SS and ask for an assessment both for her and for yourself as her carer and they will help you make the decision about the best places to look for respite.
In other words it's a bit of a minefield when there are other family members involved with different agendas but you have options and I hope you can find the right one that gives you the break that you need.
I'm sure others will be along with different views and experiences,
with best wishes from Jo