Lovely photo David! And congratulations J on the twins! Photographing a birth is challenging with one, I cannot imagine two! I hope that the staff are cooperative. It will help to bend over backwards to be nice to them without being too intrusive.
Much depends on what format you are using. The following I have used for hand-held 6x9 and 4x5 shooting with MF and LF Linhof Technikas.
If you are shooting MF, do not discount the possibility of using indirect flash, even if just to fill in the shadows and add a little light while still using a faster film/developer combination. If you are in a room with light coloured walls/ceiling you can bounce flash off the wall behind you or the ceiling above you, thus avoiding direct flash on the infant. If you balance this with available light, it will give you an extra stop of light while evening-out the contrast range (ie filling in shadows) while still almost looking like natural available light photography. This is a powerful tool for low light situations where and extra stop gives you the necessary depth of field to capture the subject(s), and you can tame the exaggerated contrast range.
I am partial to Ilford products - HP5 @ 800 in Microphen is a good combination in lower light situations as is Delta 3200 in DDX (or Microphen) @ 1000 or 1600 (but I find this last combo a little flat).
The key is to practice, practice, practice....