... If there's no precise definition of qualia, then the first task should be to find such a definition.

Definition? How about:
qualia are the language of sense (just as symbols are the language of reason). Qualia present to consciousness information which has been processed by the subconscious. Qualia are non-symbolic ("felt") information.
Consider an emotion. According to the James-Lange theory of emotion (which Ichneumonwasp cited earlier in
two of his typically excellent posts), the "feeling" (quale) of an emotion is just the subconscious (body) informing the conscious (mind) that a given situation has been evaluated as demanding a certain response (fearful, joyous, sad, angry, etc.) The quale then communicates this information from body to mind as a feeling, where it is recognized as "fear" say, and further evaluated against the symbolic (abstract) background knowledge of the subject. Emotional qualia allow us to distinguish between emotional states by differing in type and intensity and combining with each other (vague, complex, mixed feelings / emotions).
We might hesitate to call qualia "language" because they are a non-symbolic system of communication (though much of what we communicate with spoken language, even written language, is non-symbolic, dependent on context, modulation, etc.) and seem more an uncertain continuum than a discrete alphabet; however, that may only reflect our own bias. Perhaps if experience is to be meaningful, it must define qualia (otherwise how would we recognize any experience, remember it, value it?), and exist not just as phenomenal reference for linguistic concepts about experience, but as language itself.
So what if we expand the notion -- of the quale (feeling) of an emotion as a form of communication from subconscious to conscious -- to cover all qualia? Does this make sense? What is the quale of 'blue' communicating? The language of the qualia of emotions is communicated via one sense: touch (arguably, touch is two senses: external, somatic touch and internal, kinesthetic; clearly, emotional qualia exploit the kinesthetic sense to become "language"). Are color and shape (etal.) the language of visual qualia? Are pitch and volume (etal.) the language of aural qualia? Fragance the language of smell? Savor the language of taste? (i.e., "[Is] smell the language of smell? Taste the language of taste?" -- importing some odd synonyms to separate phenomena and faculty). Is their function the same: to communicate an irrational evaluation from the subconscious (level of sense and "reflex") to the conscious mind (level of thought and reason)? I don't see why not. Sounds, for example, heard through the ear, are presented to the conscious mind as significant (music? noise?) or not. As colors and shapes may become meaningful (art? familiar? unfamiliar? danger?) before awareness, smells and tastes smell like and taste like something in the conscious mind (having been associated by prior mental process).
If qualia are in fact language, and experience is a kind of discourse between sense and thought, concrete and abstract, subconscious and conscious, body and mind, then it may, in principle, be possible to replace them with another sort of language; with a symbolic language even (what's that thesis again? oh yeah, Church-Turing), as long as the same information can be communicated. This is interesting from the pov of that standby of consciousness studies, the [philosophical] p-zombie.**
Anyway, without going into more detail where I'm not too sure of my footing as it is, that's a rough sketch of one potentially useful definition of "qualia". I doubt it'll solve the HPC; but it at least suggests a compromise between the Teams in
porch's roundup, (AI) dismissing versus (HPC) embracing qualia. It would also appear to tie [human] consciousness to the [human] nervous system, phenomenally, but not functionally.
Have at it.
**
Replacing qualia, the language of experience, with a symbolic equivalent, would in effect create a p-zombie, an entity that is functionally equivalent to a being with qualia, but lacking what we normally think of as "experience". For whom, for example, "love" wouldn't be experienced as a complex scheme of interwoven feelings and evaluations and attachments and heart skipping a beat and flush in the cheeks and weakness in the knees and bonfire in the loins, but a scheme of symbols falling within the bounds of a certain definition: "I've fallen for you" equivalent to "the symbols you provoke fall within my definition of 'love'". One thing's for sure: p-zombie poetry is gonna suck (or soar, if you're sick of romantic cliches, or already a p-zombie).
ETA:
(haven't read all Dancing David's posts yet; may be covering a lot of the same ground).