Supplements - Caffeine and pre workout supplements are a possibility at helping you overcome the tiredness and it is one way that I have been using. I find 200mg of caffeine (I'm not a coffee drinker so I don't need that much, for a heavy coffee drinker you may want 400mg to feel something) and a double dose of a pre workout supplement can help to at least rev you up some more that if you did nothing. Currently I'm using the SAN V12 Magnum and it is not bad from a performance perspective but frankly tastes like a camels afterbirth!
What time you train - I have employed 2 approaches to training that seem to help. One is to get up early (in this case it may be 5am if the baby is up) and instead of going back to bed I will get up and train immediately as I find going back to bed for a few hours just makes you feel like crap. the other option is to train just before lunch so that you expel every last drop of energy and then sit down to a big lunch to help recharge your batteries. In essence however regardless of the time of your training it is more i=n relation to staying motivated to train that is more important. If you don't push yourself when your sleep deprived it will be very hard to actually work up the motivation to start again if you have had some days off.
What type of training - When you are sleep deprived there is absolutely no doubt that your strength levels suffer somewhat. Therefore I would recommend you keep the reps higher and the rest between sets shorter. By doing this you aren't allowing yourself the trap of sitting down between a set and finding it just too hard to face another set of squats for example. By doing this you are constantly moving and staying active. The other thing I have been doing is coupling exercises and staying active that way. Some of my favourites are Bench and Chins, Squats and abs (leg raises, crunches, jacknifes etc) and Military press and Barbell curls.
The key when sleep deprived is to just stay motivated (easier said then done I know) as you will be surprised how good you will feel after you have completed the workout despite the fact that you were knackered to start with.