I don't want to overstep, but you guys are doing it wrong. Your wife sleeping 12 hours doesn't sound right to me. We took shifts, but my wife was going to bed very early (9pm) and waking up around 5 or 6pm and then I'd sleep. She did this so we had a solution that worked for both of us. If your wife is not willing to do that, then you should be swapping nights at the very least.
Your wife sleeping at midnight and waking up so late is not sharing the workload at all and is frankly abusive IMO. I'd also look into postpartum issues since sleeping 12 hours is not normal.
