And there is a definite proof that these wings are not wings at all, but just shadow that looks like wings. You can read it right in the paragraph before the one you quoted.
Quote:
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The Balrog reached the bridge. Gandalf stood in the middle of the span, leaning on the staff in his left hand, but in his other hand Glamdring gleamed, cold and white. His enemy halted again, facing him, and the shadow about it reached out like two vast wings.
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That's just for completing it. Sorry to ruin your hopes, but this thread is pointless. Look at the thread
"Do Balrogs have wings?" There are pages and pages of arguments, including of course mainly the two we just quoted. Everything was already written
