You're going to have a hell of a time bonding polyethylene to anything. You can't solvent weld olefins, regular glues don't really stick to them, and heat welding doesn't work well at all with dissimilar materials.
TAP does make a glue for polyethylene, although I couldn't say how effective it is since I've never tried it myself. It looks like you'll need to flame-treat the polyethylene before bonding as well. I've had some success with flame treatment and regular contact cement in the past, but it's not a structural bond at all and doesn't stand up well to any kind of stress.
If at all possible, your best bet would really be a mechanical fastener of some sort. Barring that I'd give the TAP glue a try - being in the plastic business does lend them a certain amount of credibility.