I will be honest -- I would want to put Daniel Tiger on the TV for 30 minutes a day, at least until I could get the kids naptimes synchronized. But by the time I could get naps synchronized, the older child would probably start outgrowing the need for naps, so maybe even after I got the baby on a predictable schedule, I would still want Daniel Tiger for 30 minutes a day.

My current charges are 8mo and almost 3. Their naps don't overlap anymore, so usually we eat together. I plan (loosely) activities to be alternatively very demanding and then quiet and peaceful, like having a loud dance party or a game of hide and seek followed by some kinetic sand time. When both of them are engaged independently for 10 minutes, I will have a cup of tea while observing from the couch. We go to the park/playground on most days, so after we come back they are both way more likely to be involved in their toys without me needing to facilitate play, so I can check my email for five minutes. A nanny who has experience with a baby and a toddler will know how to carve small breaks, unless you expect her to do a lot of household tasks as well or actively engage the children every second they are awake.

I am a nanny for two children 3.5, and 7mo. Both kids nap form 1:15/30-3:30, baby will nap longer till 4/4:30 some days . i always wake the 3 year old before 3:30 bc otherwise it affects her bedtime. Are you looking for a nanny, or a nanny/housekeeper? If your looking for a nanny, whos job is to watch the kids, clean up after meals, and do some laundry here and there then you should expect all of that to be done. I also tidy the toys, and do about 20min of organizing most days, and then I have a 90min-2hr break.

I take the kids to story-time, gymnastics, the park, play dates, preschool, doctors apt, museums etc.
I set up train tracks, build pillow forts, play play dough, color, chalk, plant plants, make volcanoes, sing, dance, do counting, colors, letters, numbers, shapes, sounds, body parts ...and otherwise engage and teach both children.
I DO NOT play with them ALL day long, they have plenty of independent play between activities, and outings. Your nanny is not a circus animal, it is not our job to constantly be in the kids face, if you agree with this then she will have down time while the kids play, and while they sleep. I have had jobs where there are more than 2 kids and I still get a break. If you are playing with the kids all day then you too should encourage independent play, you will be a better parent with the down-time.