On a level that had nothing to do with the problem at hand, he was fascinated by the outing and that he was having it with a former angel. There weren't many of those, former angels.
On the level that mattered, he was very focused.
... a better solution.
To find out what, indeed, the make-up was of the people in this place, the Doctor had a few ideas.
"Right. Trauma ward is iffy. Unless..."
The Doctor held up his right pointer finger and ducked into a doorway. When he returned, he was wearing a long white labcoat. He did not have a name badge, the way most people in the hospital did, but he did have psychic paper in his pocket, and that was just as good. He removed that, and clipped it to the place the badge should go.
Anyone seeing it would see what they wished to see: credentials.
"Trauma ward. Yes. On we press. I'd very much like to see some x-rays and lab results."