I have been working on this. A lot of the eighties (and nineties) guys used Marshall 2204 JCM800's which is high gain, all tube even order harmonic distortion. They 'structure' that more organic gain with pedals that create harsher square wave clipping like the SD1, MXR Distortion Plus, Proco Rat, or the Boss DS1.
My approximation, which is maybe 85% there, is feeding a high quality OD (in my case a J. Rockett Tim Pierce OD) with a Boss DS-1 into the clean channel on a Fender. I also use a Boss CH1 Super Chorus set dry and slow and a mid length delay set a little on the dry side. This is for practice levels.
I have a JCM800, in some ways it's actually trickier getting the right sound because it doesn't like the chorus and delay in front of the amp gain so much and it has a finicky effects loop. I'm guessing that Jake's recorded sound had the mod effects added off of the recording board.
I don't know what your amp gain characteristics are like but those Boss distortion or OD pedals can be had for cheap. My DS-1 cost me $36 off of fleabay a few weeks ago. So maybe a Marshall in a box type pedal like a Carl Martin Plexi-whatever he calls it with a Boss OD/SD/DS-1 in front?
My signal chain: guitar -> Boss DS-1 -> J.Rockett Tim Pierce -> Boss CH1-> Boss DS5 -> clean amp. I'm just working with what I've got but it more or less does the trick. If I was playing with a band I'd work on setting up the Marshall.
Tone purists will probably turn their nose up at the idea of using the Fender but I'm pretty much just using it to make sound louder in this case, not too color the sound.
Gear has its importance but its not that...important, any pro knows that.
What happened in the late 70s is that Eddie (Van Halen, of course!) introduced a totally new concept & way of playing. Most people when you say Van Halen straight away think of "tapping" but honestly that's only about 10%, the surface, as his playing is much more complex than that.
The impact Eddie had was so devastating that everyone was actually forced(!) to follow his trail otherwise they would've been completely left out, and the two main aspects of his straight playing, which set him apart from everyone else, were his vibrato, much slower & wider than his predecessors's who up to that point were all influenced by either BB King or Albert King, and also the way he attacked the strings with his right hand, whether playing lead or chords.
Inside-outside-inside licks (which he got from Allan Holdsworth, one of his main influences) are part of his phrasing, so we'll leave that out of the equation as it has everything to do with personality but nothing to do with actual sound.
You can plug anything into anything and if you implement those two aspects of Eddie's playing, of course your fingers will have to fly across the fretboard, then you'll straight away are in 80's SoCal Shredland, and Jake E Lee's of course! then you add a good guitar, a raunchy amp set on heavy crunch but NOT distortion(!) and any proven good OD/Distortion pedal in between (i.e. Boss SD-1, DS-1, ProCo Rat, Ibanez TS, etc. etc.) and that's the recipe; plus Chorus & Delay sprinkled on top obviously, both of which Eddie actually used very sparingly otherwise he would've ended sounding like his clones, which incidentally he did with Van Hagar....shame.
Let's not forget that sound/Tone is in the fingers(!) those cats used a variety of gear and always ended up sounding the awesome same! So what are you gonna do, plug into e-v-e-r-y-t-h-i-n-g they've ever used?????????
Famously, Charlie Parker (another cat who could actually play...) one day pawned his sax in order to score some smack, but they had a gig in the evening(?) and all that Dizzie Gillespie & another member of his band were able to come up with was a plastic, almost toy sax(!). No problem, that's what Mr. Parker played that night, and as soon as he started blowing everything was forgotten and nobody noticed any difference....
Long may we all wail!!!!!
Jakes' Ozzy and Badlands sounds are quite different. His earlier rigs employed modded amps as was the fashion, but for Badlands the stripped back rock and roll sound of that band was simply JCM 800s with a Boss OD and DDL pedals in front of the amp.
I got to see him with Ozzy and Badlands and enjoyed his great showmanship and unique style - something of a challenge at that time with all the LA guitarists on the scene. His Badlands tone is so dry and up front in the mix compared to the wash of reverb more common at that time. Plus he employed the single coil pickup a lot in his tone. He definitely has a sound all his own despite using such utilitarian gear.