lmsAPI Functionality in HTML5 Output

I don't have much experience with Moodle, so I'm not sure what the problem would be.  Are you publishing to SCORM or AICC?  I've only tested my code with those tracking types.

The only other thing I could think of is that the lmsAPI is not the 'parent' object, but is instead up a parent level or two.  That's when I would include the findLMSAPI function I wrote in the comment above instead, since it recursively looks up one level at a time, trying to find the lmsAPI.  In which case, you would replace:

var lmsAPI = parent;

with:

var lmsAPI = findLMSAPI(this);

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If this still isn't working, I would start using Developer Tools to determine the root of the problem.