Google Maps shows that Daytona International Speedway is about 60 miles or about 1 hour driving time (without traffic problems), then KSC to Ft. Lauderdale is about 200 miles and 3 hours driving time (without traffic problems). So total, that's 4 hours just for the driving, at a minimum.
How long do you need to see the speedway and KSC? That's something you have to decide.
Will you want to eat lunch? dinner?
Sounds like it would be a fully scheduled, long day.
It is doable, although of course your time at each attraction would probably be a bit rushed. I checked the Daytona tour schedule and you should be able to jump on a 90 minute tour at 9:30 AM. Figure you leave Daytona by 11:45, get to KSC by 1, and that should give you a decent amount of time there. I'm not sure how the KSC VIC hours might vary, but right now the place closes a 6. You could probably be down in Laudedale by 9ish.
What is included in the Daytona tour depends on what's going on on the track. If they're testing they won't take the tram on the track surface and may be otherwise limited. You might want to check ahead for the day you'll be there.
KSC has an nice museum and activites right at the main attraction and also bus tours out to very worth while secondary attractions and/or sights. The only time I was there recently we took the Cape tour, because my Dad spent time there back in the day and wanted to see the old place.Edited: 21 September 2016, 09:46
To really get the full effect of KSC you need pretty much a whole day.