Your chances of landing an acute care job in the Chicago area as an ADN new grad are slim for quite a number of reasons, already explained above. Something to consider is that since you're almost (but not yet) complete in your 1st Bachelors, you might consider completing RN prerequisites and transitioning into a BSN program and then complete both Bachelors Degrees at the same time (effectively as a double major). Otherwise I would suggest completing your current program, get your prerequisites done at basically the same time, and apply to both ABSN and ELM nursing programs
. These programs will generally take about a year to complete as both are "accelerated" compared to a traditional program. The local ABSN program (to me) still is 4 semesters but they run straight through Summer so you start in the Fall and complete the program at the end of Fall Semester the next year. It works out to something like 14 or 15 months instead of 22. The downside of doing things this way is that there's a very real chance your student loan/grant/etc funding sources will dry up if you're pursuing a 2nd degree (or a lesser if you go ADN) as you'll already have a Bachelors. There may be different funding options for Masters-level programs but I have little info on those as I haven't looked into that.
As to taking the NCLEX, it's the same exam anywhere you take it. What matters is passing the exam and having the results sent to the state where you've applied for an RN license. I applied for a license in California and took the NCLEX at a site that happened to be less than 20 miles from home. I could have gone to Hawaii, Florida, or wherever else on vacation and taken the NCLEX locally there (if there's a Pearson Vue site that's appropriate for it) and the result would have been the same.