I hesitate to recommend Italian to a Chicagoan travelling as we have good Italian options here, however . . .
Right next to Delmonico (inside the Venetian) is Valentino. I have had a number of great meals there and the service and atmosphere might be better than Delmonico. If Italian is OK with you, this is a good option.
Across the street at Treasure Island, err, I mean TI (they have rebranded it to appeal to a youth movement that probably doesn't know who Robert Louis Stevenson is) are a couple of popular places for us - Buccaneer Bay Club and Francesco's (another Italian, but with somewhat of a focus on seafood so maybe not for you). I have to say my last visit to Buccaneer Bay Club disappointed, but it was Super Bowl and I sensed the casino hosts had overloaded them with forced reservations they probably wouldn't have taken if they'd had a choice. Service suffered as a result. My only "bad" experience there.
I think next door to TI you can hit a real home run for the romantic and fine dining aspect at Renoir in the Mirage. Simply outstanding - service, food, and decor that includes original Renoir art.
Down the strip a bit at Bellagio are a number of good choices, too. I haven't yet had the chance to try Le Cirque there but have dined at it's sister, Circo, with good results. I also haven't dined at Picasso there but trust the reports of friends who have; nothing but raves. One of these 2 is on my must-try for my next LV trip.
Of all of these, I would most strongly suggest Renoir since you do mention romantic. It's probably the smallest and most intimate of any of these rooms. After dinner, depending on the day, there's usually a piano player/singer in the small lounge in between the Mirage restaurants. Last time there the guy was really very good and the 4 of us enjoyed a couple of hours relaxing after dinner with a couple of drinks and some great music, talking to the pianist (one of our group was a prof. musician). Perhaps one of the best renditions I've heard of "Nature Boy" outside of Nat King Cole.