Agonal respirations was also a good thought but I think Cheyene Stokes sounds more likely. I've seen it a few times with head injuries and in strokes. If you've ever seen it it's very distinctive.
Agonal respirations generally are only the last few breaths before death. Maybe 60 seconds or so. According to the report Lincoln lived for 30 minutes after this pattern of breathing began.
Biot's breathing is generally lots of short equal breaths followed by apnea then repeating. Also a good answer but from the description it doesn't sound like The president was taking equal breaths during this period but more very short inspirations with long exhalations. This matches up with what I've seen in the field with Cheyen-Stokes.
Kussmaul's Breathing is just very very fast Rapid breathing with I believe lots of gasping. I've never seen this in the field but again it doesn't match the report.
This concludes emtlifes very first semi educational sticker contest

I'd be interested to hear how this type of injury would have been treated by modern medics. Any ideas?