Hi @HillaryA,
Welcome to L.A.! 
Within walking distance of L.A. Live? (warning lots of tourist trap / national chain mediocre food there.) Walking distance-wise:
- Broken Spanish - Some excellent, interesting dishes. Posted some pics and a review in this thread.
- Fogo de Chao - Brazilian Churrascaria (2 blocks north).
These are technically walkable, but I’d recommend a super-short Uber / Taxi ride instead (just a few blocks further away):
- Shibumi - Ask to sit in front of Chef David. Excellent Kappo cuisine.
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B.S. Taqueria - Chef Ray Garcia’s more informal sister restaurant to Broken Spanish. Some delicious gourmet Tacos!
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Langer’s Deli - The best Pastrami in the United States. Get the classic Pastrami on Rye (add your own Mustard). Skip the #19.
Splurge Meal - Does it have to be within walking distance of L.A. Live? Or anywhere in L.A.? That will help us give you better suggestions. But if it’s anywhere in L.A., then…
- Mori Sushi - Ask to sit in front of Maru-san (Chef-Owner), and go for the Omakase course.
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Shunji - Ask to sit in front of Shunji-san (Chef-Owner) and call ahead for a White Truffle course with the Omakase.
- Providence - Chef Michael Cimarusti’s flagship restaurant. New American / Tasting Menu.
There are plenty more suggestions, but hope this helps!