Ok, I found the problem (I think). LyX makes an error importing the raw LaTeX. (I pasted your code into an algorithm the first time, which is why it worked for me.)
-----------------------------------------------------
LaTeX Error: Something's wrong--perhaps a missing item.
...uclid} \begin{algorithmic}{[}1] \Procedure
                                                {Euclid}{$a,b$}\Comment{The

Note the braces around the left bracket in the optional argument [1]. You can correct this by deleting [1], then clicking on the ERT inset containing \begin{algorithmic} and inserting the [1] into the inset. Oddly enough, if you type a left bracket anywhere inside the float, LyX surrounds it with braces. I don't know why it does that, but I can't imagine it's a good idea.

Seems to me we might have two separate bugs here: (1) not putting the optional argument [1] inside the ERT with the \begin{algorithmic} command; and (2) putting braces around all left brackets. You might want to file bug reports on these.

/Paul

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