There are two buses to Nottingham that go past the Days Inn. Well, just around the corner on Nottingham Road. If you search for Trent Barton buses and either the i4 or the Indigo will get you to Broadmarsh bus station in Nottingham in just over an hour. The i4 is the quicker of the two.
It isn't far to the train station in Derby from the Days Inn, but if you don't know the area it can appear to be a bit difficult by just going on Google maps. Turn left out of the Days Inn, and bear left at the Pentagon Island. Cross over Nottingham Road and follow the path round to the other side of the island heading in the direction of the ci centre. The path bears left past the bus depot, then you will see the Smithfield pub next to the river. Walk past it, and then cross over the river at the first bridge. The bear left and over the next bridge in front of you. You'll then have a road to your right, keep on this path and the station is about 200 yards to the centre right direction. All in all about a 15 minute walk from the Days Inn.
Whichever option you choose you will have a 15 minute walk in Nottm to Trent Bridge. I think though, there are some major works at Nottm station this summer with lots of train cancellations, so might want to research that before deciding what to do.