I was going to ask you if you were my DH but you say you do chores to get out of childcare so you can't be! 
Firstly why are you up? I ask because I am awake breastfeeding but I am guilty of googling between feeds to my own detriment sleep-wise.
Is this your downtime?
Are you insomniac or depressed?
Are you on a night shift?
Are you doing this post while wife is asleep as no chance tomorrow?
Unless a decent reason you need to sleep now if exhausted - is your 2 year old not sleeping through yet?
How many kids do you have? Is your wife SAHM? Has she got PND or dd2 SN?
Bottom line - if she is knackered I am not surprised as childcare is knackering - for some a job is ligt relief in comparison. Add depression and guilt to the mix and I can see why she is sick.
You know this as you find housework preferable to childcare (not saying babysitting as it is your child too) - A stand-up comedian used to joke about sitting in his car an extra ten mins mentally preparing himself for walking through the door, in other words...hiding.
It all depends on the circumstances. Working full-time then being expected to do ALL the chores, maintenance, cooking, childcare etc AND acting as carer for your wife is shit...but doing 50% is fair if she has had DD all day wit no respite, lunch break, toilet break etc esp if there are SN levels or sleep problems.
You need to think of support mechanisms...is there family who could help out? is a GP referral needed? Is Surestart or whatever it is called for helping out a possibility? Is a cleaner for two hours a week an option?
Exhaustion itself is exhausting.
I do understand.
The house won't clean itself. The kids cannot self-soothe or entertain and it is not called the terrible twos for nothing.
You could try each doing one hour on one hour off, one doing chores, the other DD and rotate it each time then when DD finally down, both have off time.
One lie-in each Sat / Sun - unless she is on night shifts - two years of ebf and cosleeping is draining.
Would need more info to help you more aside from you sound like you have her back and care but also are running on empty too.
It gets better between 7 and 10yrs do not put head in oven now, tis only 5 years away