Well, taking the headphone out and running that into the input jack set at "LINE" might (repeat, MIGHT) work. BUT you need to get an adapter that takes the (2) channels off the headphone jack (you know, 'L' and 'R'), and feed them into (2) seperate input channels on the recorder, one panned all the way to the Left, the other panned all the way to the Right. You can get that adapter at Radio Shack; it has a male 1/4" TRS (tip-ring-sleeve, stereo) plug at one end, and then (2) TS (tip-sleeve, mono) 1/4" plugs on the other. OK, maybe RCA plugs and you'll have to get RCA-to-1/4" adapters, no biggie, always handy anyway. This way, you'll be taking the (2) outs to the recorder properly. If you try to use it mono, you'll be having issues with any stereo effects processing done inside your H&K (if any). And using a mono plug on the phones' jack may not be cool to the amps' circuit. Be sure to set the level to "LINE" on the recorder. This will keep the relatively "hot" level of the headphone jack from overloading the recorder's inputs.
Something else to consider: The internal speaker ( and, to a lesser extent, the headphones) acts as a giant "lowpass filter" that cuts out a lot of the high frequency "grunge" and fizziness" that the circuitry is actually generating, you just can't hear it. With a 'direct connection' eliminating that, you're gonna hear a different sound. Be prepared to roll off a LOT of the top end at the channel EQ to get this to work.
And if that doesn't work, get a POD!!!

'Headphone-out' passive DI

Thanks for your advice. This is very helpful.

I would love to build a simple passive 'headphone-out' DI box with a pot or two to attenuate a headphone-level source to line-level output. This would be an incredibly useful little tool for grabbing audio from miscellaneous audio gadgets. If anyone could help me with schematics or advice I would be incredibly grateful. I don't know much about circuitry but I can solder half decently.

Actually you can just get a 1/4" trs to 1/8"trs adapter or cable(preferrably a cable). Headphone out to line in.

You can buy a instrument to usb cable now. I've never used one, nor would I put any trust in one but if this is just a hobby for you there a lot of cheap and free options.

Otherwise work out a budget and be prepared to spend a little money. It doesn't have to be a lot.

Buy a mic, then construct a giant fort around your amp. I have the same problem with a 1/2 stack so I mic in up then cover the bastard in heavy blankets.