No. Cologne is in Nordrhein Westfalen... North Rhenan....???
The question is, how much time you have... Tickets like the Germany ticket or the cheaper state tickets just let you go with the regional trains, not the speed trains. From Frankfurt or Stuttgart to e.g. Heidelberg that might cost you an extra hour, often a lot less. From Cologne the difference it takes less than 2 hours with the ICE but nearly 5 with the regional trains with extra train changes.
You would go through atleast 3 different German states, so probably in case you would prefer the cheaper trains you should take the Germany ticket we talked about. Some have put links for it in this thread,
If you go alone it could be even cheaper booking online the normal fare. You have to look into the schedule and the booking page of the Deutsche Bahn as different connections as well as different tickets for it - some are not refundable or bound to taking that train - cost differently.
Cheapest is probaly Flixbus. Booked online it costs from 13,99 Eur, but the latest possible trip to Heidelberg starts 8:30 and arrives 13:10. There is 2 busses to Mannheim at 10:30 and 13:25 arriving in Mannheim at 2 pm and 4 pm 30. Problem with the busses is that they often are late.