Sorry to hear you had to adjust your plans. A few comments, suggestions, observations, etc; I'll start with some related to your questions:
1. Certainly the Children's Museum should be near the top of your things to do list. Beyond that there is the Zoo (weather might cooperate for that) as well as a few museums in the downtown area. I'd check to see what the Indiana State museum has in the way of special exhibits as sometimes these will have interests for children; also it has an IMAX theater often have 2 rotating shows going on (usually one is nature oriented & the other an "entertainment" type movie). Don't really have any other ideas for 5 year olds off the top of my head., will try to come up with some & post another response (did I miss a gender reference?).
2. Even though your plans have fallen through, I'd still encourage you to spend time in Bloomington. To me, smaller cities/towns, such as Bloomington, have a more "Indiana" feel to them than Indy - which while it certainly is an Indiana city - still has more in common with other major cities. Some others you might want to check out (most are in the general Bloomington area), include: Nashville, French Lick/West Baden, and Columbus. (I believe there is a thread in the Bloomington forums discussing these and others in more detail.)
3. A few Bloomington ideas: Wonderlab (hands on children's science museum), there are many museums (most, but not all, part of IU and would have some appeal to small children, we're not talking purely art here), walking around the IU campus, etc. Then there's *lots* of outdoor stuff, but it's a little iffy to plan on that as the weather could be nasty (or nice).
4. As for Basketball, it's likely you can find a game; but perhaps not a *pro* game. There's also multiple college teams (although IU, in Bloomington, might be hard to get a ticket for, the teams doing well this year) which will play just about any night of the week, and if your dates are over a weekend (too lazy to check), there will certainly high school games you could see. Personally, I *much* prefer college and/or high school basketball to pro. I often use the Indiana small town/high school games to describe how big basketball is in Indiana: town of maybe 5-10,000 people, high school gym will seat maybe 2,000+ and it will fill up for most games. If you could catch a game at Hinkle Fieldhouse (that's Butler University in Indianapolis), I'd say go for it; this is perhaps *the* classic/old school basketball venue in the entire state.
5. As far as great places to eat. I guessing that with a 5 year old you're not looking for 4/5 star fancy restaurants, but some more guidance would help (budget, location, cuisine, etc.). As for a "quintessential" Indiana food, I'd probably go with a breaded pork tenderloin sandwich. Plenty of places have it, some better than others. I'll try to come up with some specific suggestions later, in the mean time you might check out Urbanspoon. This is restaurant oriented site that crowdsources (much like TA) and Indianapolis & Bloomington both have the necessary critical mass of users that make the data significant.
6. Your location could also influence suggestions, where in the Indy area are you staying and/or let us know any other cities/towns you might be passing through and/or staying in (such a Bloomington).
Lastly, for what it's worth, "Hoosier" is probably a better phrase to use than "Indiana." Indiana is the Hoosier State and people from here are known as Hoosiers. As for the origin of this name, it's lost to history; there are multiple stories, but none of them has any definitive proof that it is the source.
That's all for now, maybe more later,