Hi, your cat is so pretty and what a great age she is. She has been very well cared for to get to such a great age. I have been at this point more times than I care to mention as I work in a vets and with my own fur babies too.
Before tomorrow take a clipping of her hair and using some paint ( any paint, ideally matt as it will wash off) get a couple of pawprints from her. Or use an ink pad if you have one handy.
When the vet arrives ensure you know the process is very fast, the vet and most likely a nurse too will be there to assist in the process. They clip some hair from one of her front paws and place a cannula ( needle) and tape it in place. They will ( or should, do ask them to) give her some sedation at that point so she is sleepy, then they will ask you if you are ready to proceed and they will inject the euthansia drug. Within a minute Lola will slip away for a forever sleep.
Then afterwards they will either depart and leave her with you to bury in the garden or take her with them if you decide on cremation. If you decide on garden burial, make sure you dig a hole a lot deeper than you think is necessary, esp as you have other animals and children.
There are 2 types of cremation, group or individual. Group - no ashes are returned, they are usually scattered in the cremation companies garden or a field they have permission to use and if you choose individual cremation you select a casket and get Lola's ashes back. There is no right or wrong choice, just whatever is right for you as a family.
Your children can be there or in another room, if there with you just explain that Lola is old and tired and she is going to a special place where all pets go when their bodies get too old to be in this world, and the vet is going to help that happen. If the kids are old enough they will understand and may cry, if younger some just accept it but will have loads of questions afterwards.
I wish Lola a peaceful passing tomorrow and sending all the love and comfort to your family to get through the coming days.