speechless

Both are correct. The adjective, however, doesn’t fit the structure of the clause, so it will need to be set apart by punctuation: I looked her straight in the eyes, speechless.

I don't see how you can look speechlessly.

I looked her straight in the eyes speechless/speechlessly.
Also, assuming the one with speechless is incorrect, how do I correct it?

Here are some correct versions.

I looked her straight in the eyes speechlessly.
Speechlessly, I looked her straight in the eyes.

@enoon: you're confusing look (appearance) with look (to look at).

C Bryan King@enoon: you're confusing look (appearance) with look (to look at).

If he is, so am I. I do not believe that I can look speechlessly at anyone or anything.

Well, you can.

Excerpt from Falling by John Taylor
http://www.nytimes.com/books/first/t/taylor-falling.html
"I looked at him speechlessly. He shrugged and, as the train pulled into his station, touched me on the shoulder and said, 'See you at the game.'"

C Bryan King"I looked at him speechlessly. He shrugged and, as the train pulled into his station, touched me on the shoulder and said, 'See you at the game.'"

Fine. Some people use it. I don't regard it as a major sin but, as an aging pedant, I am quietly amused by the thought of somebody looking (verb) speechlessly (adverb) at someone/something.

C Bryan KingExcerpt from Falling by John Taylorhttp://www.nytimes.com/books/first/t/taylor-falling.html "I looked at him speechlessly. He shrugged and, as the train pulled into his station, touched me on the shoulder and said, 'See you at the game.'"

I am not

enoonyou look someone straight in the eye, not in the eyes.