Been a while since I have been on the forum. After getting the bike sorted out last fall, I had been riding it here in the Upstate region of South Carolina where I relocated for the past 6 months or so without an issue.

The other day I parked it in the garage and plugged in the battery tender jr. (as it was going to be a few days until I rode). I went to go to work a few days later and turned the key, but nothing when I hit the start button. When I flipped the kill switch in either direction, the bike emitted a buzzing/humming sound from under the tank/seat area ... in the relay box.

I didn't have time to do anything other than grab the truck and head to work, so there it sat for another day or so. On Saturday, somehow, somewhere, I lost my only key.   No way of getting another made, and the barrel and key assembly isn't available from BMW any longer. I figured I would just remove the lock section and look for something on eBay, no big whoop.

So, today, I FINALLY had a day off and got into it. I pulled apart the ignition lock and within several minutes, was able to turn the switch with a screwdriver. Now for the testing. Same exact result as before when I tried. No start, no sound when ignition is on and kill switch is in center position. Flip kill switch to either side, buzzing from relays. So, I decided to siphon the tank, remove it, and open up the relay box to get a closer look-see/listen. The sound is coming from the load-shed relay I think. I didn't hear the fuel pump engaging before I removed the tank, and now I am at the head-scratching phase.

Care to weigh in?

There's plenty of others who love a good electrical problem. Keeps them going for days. 
My limited knowledge would say that you shouldn't hear anything when you switch the ignition on.
The pump only whirs when you hit the starter and stops a couple of seconds after you release the button, that's if it doesn't start.
If it starts, the pump continues to run.
Your problem is one I haven't heard of before but that will give some of them a challenge.

If you suspect the LSR then try swapping it with the horn relay. They're the same. Both are in the front/right corner.

Swapped the relays and still the same result. I guess I am going to have to pull the starter and try that. But, no more time today, and 4am comes awfully early! Biggest drag is that I have missed some great riding weather over the past 10 days or so ... 

Oh well, I guess that is the price for having a 30 year-old bike. Maybe it's time for a 'Busa ...

Have look at the troubleshooting page off the Portal Tab.

No sound from fuel pump/relay or the starter relay would indicate a power problem somewhere.

What voltage is your battery reading? The fact that its been on a trickle charger doen't guarrantee a good battery.
Are all the fuses ok?

You could test the start/kill switch functions with a meter first.
Sounds a bit sus. :suspect:  It may just need a good clean and deoxit.

Also worth checking that the connectors for the FICU (under seat) and the ICU (under tank) are clean and secure. Wouldn't be the first time either of these have been the culprit.

Could also be an earth connection problem. They need to be kept corrosion free. Gear box, under tank and starter.

I don't think the starter is your problem tho as you also aren't getting fuel pump.

Big fun with a small toolkit. Removed the ignition switch, tank, compute, battery, and (eventually) the starter. There was some corrosion on the positive terminal of the starter connection, but I figured why not remove the whole thing and clean it up. The rubber counterweights were kinda gummy, so mayhaps I need a new set of those too while I am in there?

go easy pulling things apart. You need to start with simple 'likely' things first. More often than not you'll find its just a loose/corroded connection. Also you may not then know what the fix was when you put it all back together.

Really important is to find out the condition of your battery. Buy a cheap ($10) digital multi meter (DMM). All sorts of weird and strange things can happen when your battery gets low. You said its been charged but that's no guarantee its good.

Cleaning earth contacts, start/kill sw. and ignition sw. would be next, or at least testing those switches with a meter first.
If its not the battery, my next guess would be the start/kill sw.

The 'monkey nuts' (rubber counter weights) are cheap enough to replace if needed, but what you've removed is the alternator. The starter is on the left. Check brushes for wear. Specs are in the manual. Worn starter brushes can cause strange problems.

Not a lot to do to the alternator unless you pull it apart and refurbish it and there is nothing to suggest you need do this so leave it alone. Just check the brushes and clean the contact on the voltage regulator. Worn brushes here or dud voltage regulator will result in poor charge voltage while running. Make sure you replace the alternator connector before starting the engine