Whilst I agree with the PPs who are pointing out that your youngest is only 5, and needs to feel secure [and I agree with them that it's up to you to make her feel secure in this scenario], I also feel that your youngest has too much power in this situation, and that you are the adult and need to be in control.
And I think you need to help your 5 yr old get over the fear and insecurity.
But also, I feel a 5 yr old shouldn't be up until 10 every night.
So I'd work on getting a better sleep pattern re-established.
Personally, I'd put your plan of exercising while they're in bed on hold for a while, especially if that'll help her bed-time routine.
Perhaps do a bit of exercise while they're still up and can come in with you (maybe have some screen time sitting at the side while you do your thing). Perhaps do 10 minutes while they eat tea together (with the video monitor suggested by others).
Then, once the routine is re-established, you can start to think about what you want to do in the evenings. This is YOUR time.
Having said that, I would never have left a child home alone, and this sounds a bit like your set up. But perhaps the geography of your home isn't like that?
If there's a fire and you're out on your cross-trainer (or whatever it is), would you have time to notice it/get back to the children/get them out in time?