Normal is12.5 or better.
If you charged it and it's not even twelve,,
Put a battree in it.
That is The most common problem with this bike. The ignition requires a decent battery to
Crank the start
And
Have enough Oomph to fire the plug.


The fuse you replaced
Blown?Not blown?
Relays and solenoids
Click.
If you hear a Rattling Clicking, that is the solenoid with
Not enough battery to Engage .
The coil that creates the electromagnet can't stay energized because the battery is kaput

I am going to try and offer some info that I hope will be useful to you.  Although the Savage is a simple beast, it’s electrical system seems overly complicated to me.  I will try to give you a better idea of how the current flows.  

The first piece of advice I offer is about nomenclature.  Your vid was nice, it helps us see your problem.  Correct nomenclature is very important.  I suggest when you want to verbally identify a part, go to one of the on-line parts distributors and find the illustrated parts breakdown for your LS650.  Find the part in question and use the nomenclature they provide.  The little black box you replaced is not a fuse, it is a “Decompression Control Unit”.  Here’s a link to Babbitts Suzuki Parts House.  It’s a good tool.

https://www.suzukipartshouse.com/oemparts/c/suzuki_motorcycle/parts

The second piece of advice I want to share is that a battery that only has 11.7 volts of potential is “DEAD”.  I concur with JOG & Versy.  A fully charged battery must have 12.7 volts of potential.  That being said, the fact that you can jump directly to your starter and get the starter to crank the engine tells me that you should at least get a little action when you hit the starter button.  So, aside you’re your tired battery, something else isn’t right.

The electrical power for all of your components runs through two fuses.  Those fuses are identified on the wiring diagram as “Fuse 1” and “Fuse 2”.  They are both 20 amp fuses.  They are located under your seat in a small white plastic box.  I can see the box in your video.  This is the fuse box in question.

You want to make sure that both fuses are good.  If fuse 2 is burnt out, it will cause exactly the problem you are experiencing, decompression controller clicks but starter does not operate.  So, start by checking your fuses.  If one is burnt out, then replace it and see what happens.  Hopefully it doesn’t just burn out again when you hit the power switch.

There are two fuses and they both provide power for your starting system.  Power flows from your battery to both fuse 1 and fuse 2.

FUSE 1

Power flows through fuse 1 to the ignition switch (the switch under the left side of your seat).  When you turn on the ignition switch, most of the systems are hotted up.  This is the flow path for fuse 1.

Power flows from the ignition switch to the side stand relay.  It is located under the fuel tank, on the left side, just below the ignition coil.  This is the side stand relay.

If the system is working correctly, the orange/black wire is hot when the ignition switch is turned on, and the orange/white wire gets hotted up (connected to the orange/black wire via internal contacts in the switch) when the switch is in the “run” position.

The connector plug for the right-hand handlebar control switch assembly is under the fuel tank on the right side.  This is the connector in question.

When the starter button switch is pressed the internal contacts connect the orange/white wire to the yellow/green wire.  The yellow/green wire should be hotted up when the start button is depressed.  You can find the wires in the connector assembly under the tank, the one I described earlier.  As the picture shows, its big, has lots of wires, and is partially covered with a vinyl boot.

From the starter button switch, power flows via the yellow/green wire to the clutch lever switch on the left-side handlebar control switch assembly.  This is the clutch lever switch.

From the clutch lever switch, power flows via the yellow/green wire (out) to the decompression controller, where the power starts a timer system that operates two separate relays within the controller.  One relay controls the decompression solenoid, and the other relay controls the starter motor relay.  As you already know, this is the decompression controller.

Since your main concern right now is getting the starter turning, I’m not gonna go into the decompression solenoid.  If you have battery power to the decompression controller red/white terminal, but still get no audible indication of the starting motor relay relay activating, check to see if the starting motor relay gets hotted up when the starter button switch is pressed.

Power should flow from the decompression controller to the starting motor relay via the yellow/black wire.  There is a plug under the seat that connects the decompression controller to the starting motor relay.  This is what the plug looks like.  See the yellow/black wire?  It must get hotted up when you press the starter button.