Give that 15 minute warning but add a 10 minute warning, and a five minute warning.
At 15 minutes, the girls must pick up toys. At 10 minutes, the guest has to get on shoes, gather whatever she brought, etc. Have the girls IN the living room near the front door from 10 minutes onward (they might need more time than that, too). Do not let them keep running back to the host girl's bedroom or other rooms -- the play date moves to the front door area no later than 10 minutes until pickup time! If they forget something then you, not they, go and get it.
If your child is the one doing the leaving, be sure you have somewhere else to be: "Dad expects us home in 10 minutes" or "We have to be at X right away." (Be sure it's true -- the kids will very quickly learn if it's a fake-out.) That tactic also should telegraph to the nice but chatty mom that you also do not have time to chat.
If you are the host, and girl and mom are dawdling, turn it around: "Dad will be home in 10 minutes from now and we are leaving as soon as he gets here go to dinner" is always motivating...but if it's not true that night, then announce loudly and firmly, "Girls, Daughter has to start X (homework, helping with dinner, picking up room, whatever) in five minutes and can't start while you're here, Friend. We'll see you next time!"
Always have something your kid must start doing, someone who is about to arrive home shortly, somewhere you must take your child.
I never liked the hide-and-seek "just another minute!" stuff at the end of play dates either, but bringing the play date up into the living room 15 minutes ahead of pickup and even closing doors to the rest of the house to signal that those areas are now done -- that really helped.
When you are the one picking up -- You do need the other mom on your side here -- don't be afraid to tell her that "Daughter is having a hard time leaving play dates politely and promptly so when I come to pick her up, I'll be hustling her out the door -- it's not personal, we're just trying to teach her it's polite to leave nicely." Also try phoning the other mom 20 minutes before playdate's end and asking her specifically to give your child a 15 minute warning! Most parents are happy to do it.
Yes, if your child throws a fit or just is very balky and difficult at the end of a play date, give her one warning, but do take away something she values if she keeps it up. It's basic good manners to learn to leave a play date or let a guest leave nicely. It is learned and you can teach it -- hang in there, mom , and discuss this with the other mom!