Jack started with the Time Lord, knowing that he was skiddish about needles. So doing what he needed to, Jack took enough blood, pulling out the needle and placing a cotton ball on the puncture wound. "There. All done." Jack told him before placing a kiss on the Doctor's forehead. He then got a new needle and syringe. Arwen held out her arm to Jack so he could take some of her blood. Jack did so and she wasn't phased by any of this. After that was done, she too put a cotton ball on top of her puncture wound. 
The immortal cleaned up everything before going over to the scanners to have everything tested and see what the TARDIS comes up with.