Seriously, give your kids a snack. The idea that you get home at 5:45 and will have a real meal on the table by 6:10 is ludicrous.
If you spent 15 minutes giving them the undivided attention they want while sitting down with them over a snack, you could then aim to have dinner on the table by 7 and save your sanity.
This is an example of the "hurrieder you go, the behinder you get."
+1, I have a similar schedule-- home by 5:45 kids starving. Ijust give them a healthy snack to tide them over and then we eat about 6:30-7. Kids in bed by 8-8:30.
Honestly, if your kids go to bed at 7:00 and you are only getting home at 5:45, you sound like you are stressing out over an unrealistic schedule... even pushing bed time back to 7:30 might give you a bit more room to fit things in or as a PP said have the nanny get dinner started.
It sounds like your kids are young and don't have homework yet--- so take it from me, who had to learn the hard way...your schedule will have to change at some piont (kids go to bed later or you get home eariler or the nanny cooks) in order to also fit in homework.