We had a stretch where I was working evening (getting home anywhere from 11:30pm to 1 or 2 am) or night shift (7-7), and my husband was on a submarine. He worked 12-16 hr days, often had to go in at random times, and then left on patrols for a few months. We settled on having a nanny, and it was definitely the right choice for us. We had 2 kids though, so it didn't make as much a money difference vs a daycare center. We went through care.com, and had two excellent nannies over the course of 3 years.
We paid $17/hr, with a guaranteed 30 hrs/week, in a MCOL area. We also gave PTO for sick/vacation and paid holidays, or time and a half if she needed to work on the holiday. The benefits, decent pay, and guaranteed hours help keep good nannies around. With taxes and paying the payroll agency, it cost us ~$2600/month. You can pay for background checks through care.com.