People take a strange view when it comes to "legal" things, people in general know very little about them and so don't want to get involved in case there is any come back.
If they're not prepared to do it jointly I'd ask each of them if they had any objection to you asking the other one to do it alone? It may be that they have different views and don't want to do it together, if they still refuse then it may be worth pointing out to them that if you pick a friend or a relative to do it that this person will have all the say over your health and welfare and you finances, their status as your children, should anything happen to you will be overridden by the LPA holder.
This person will have the total right, should they want to sell your house and all its contents on ebay and do things with the money that aren't what your children want.
Jen's suggestion that you use a professional, lawyer or whatever is fair enough but it will cost...big time so if the kids don't mind seeing their inheritance getting eaten up at the rate of several hundred pound an hour by a solicitor then that's their choice.
Do they realise that if no one holds an LPA and you were to lose capacity then all you bank accounts could be frozen? If you were deemed to have no capacity and no one has a LPA then they have no rights to do anything, no access to your medical records, no right to be consulted about your care, no right to use your money to pay your bills, the fact they're your children will give them no more right to a say as to what happens in your life than I do?
If you can't find a way to make them work together and they won't do it individually either then suggest to them an (unacceptable) alternative person you might pick, the idea that mad auntie Doris in Auchtermuchty being the person responsible for managing your future health and their future inheritance might focus their minds
Whilst everyone has the right not to do something I think your children's view is very short sighted, I'm not sure they truly understand how much it could cost in the future if they don't have an LPA and how much their hands will be tied.
K