Ask if they can apply EMLA cream well in advance (you may then have to wait quite a while for it to start working though) in several places. The key is lots of distraction, something new, like a book with noisy buttons she can push with the arm which is not used. Have something sweet for her to suck as that is expected to reduce the pain sensation.
I find sitting dc on lap facing you with you holding the book behind your shoulder for her to see/push buttons. I would not let her see the blood taking as many kids find that very frightening. Tell her a story in the meantime or tell her to push on the buttons, basically distract her and have her focusing on you/book. I always give a small present directly afterwards. I have had to do this many many times and most of the times dc didn't make a sound/no tears, so it doesn't need to be traumatic for dc (probably is for you though).