John,
Long time no see. Welcome back.
Smar's suggestion of Annunciation is certainly more than worth considering - the parish's reputation and that of its pastor are widespread and deservedly so, from everything I have ever heard. But, just in case you don't want to go that far (I'd guess it to be about 30 miles) ...
It seems to me (relying on fond memories of my very favorite city) that SS. Volodymyr & Olha (Ukrainian) is the closest Byzantine parish to the Holiday Inn, probably not more than 5 miles (about the same distance as the Cathedral). I don't know the schedule for Divine Liturgy at SS V&O, but knowing what I do of the parish, I suspect that it is likely not served in English (which I think is what Lawrence is suggesting as well). I agree with him that the churches he's mentioned are all beautiful and worth seeing.
Ss. Peter & Paul (Romanian) Mission
(Father Sergiu Cornea, Administrator)
3107 Fullerton Ave, Chicago, IL 60647
Tel.: 630-896-4807
SS. Peter & Paul is about 6 or 7 miles from the hotel; it used to have a Sunday Divine Liturgy at 9:30 AM (in English and Slavonic), but I have no idea if that is still accurate. It's located at the site of what was formerly the only (and, now, regretably, suppressed) Bielarussian Catholic parish in the US. They don't have a website, as far as I know, but you can see a picture of it on St. Michael's Russian Catholic (NYC) site. It's a small church and lacks the magnificence of the established parishes, but it has a beauty of its own.
SS. Peter & Paul Romanian (scroll down to Christ the Redeemer) [
stmichaelruscath.org]
I think I remember that Michael (Coalesco) has been there more recently and may have some info to add.
There's a Melkite parish in North Lake, about 15 miles from the hotel; Liturgy is in English and Arabic, but it's at 11:00am, which may be later than you'd like - you didn't say what time the flight is.
I haven't been in the Windy City for about 5 years now and don't have a clue what taxi rates are like; hopefully, Amado (Amadeus) or Lawrence will chime in with some info on that. With that caveat, here's another option:
Nativity of the Blessed Virgin Mary (Ukrainian)
8530 West 131st Street, Palos Park, IL 60464
tel: 708-361-8876
Liturgy: Sat 5pm (?lang), Sun 8:45am (Eng)
approx. 25 miles
There are several options for Eastern Catholic Liturgies in the non-Byzantine Churches; there are Chaldean, Syro-Malankarese, Syro-Malabarese, and Ethiopian parishes within about 10 miles of the hotel and a Maronite parish about 15 miles away. I'll see what I can find out about languages/times, if you are interested in any of those.
Many years,
Neil