For the time you'll spend in Beech Grove, check-out the University of Indianapolis Ruth Lilly Fitness Center. They have had public hours in the past and may still, and it is very close to Beech Grove. Beech Grove is located entirely within Indianapolis, and if you have time I highly recomend the Natatorium at IUPUI. It's a little farther away from Beech Grove (20 min drive), but it's a world class facility and a cool place to swim. And if all else fails send me a message, and I will give you some guest passes I have for the Baxter YMCA (10 min from Beech Grove).
As for Seymour...good luck. I don't know it very well except to know there isn't much there.
I am unsure about Seymour... but here in Columbus (just a short drive away) there are a couple of options:
1. Total Fitness. They are a health club and I'm sure they offer a day pass.
2. the Foundation For Youth. Not sure how long you'll be here, but generally you have to sign up for a "membership" ($25) and then buy "visits" (about $3 each) If you're just going to be here for a few days, let me know and I'll let you use my card. I have some visits left and I'm not going to use them before I move to China at the end of the month!