Work and energy problem with friction. Very confused

You've left quite the word salad here, but I'll try to answer your question as best I can nonetheless. I'll start by showing how a way to solve it, and then why your method didn't work:

At the top of the slope, the body has gravitational potential energy. As the body falls down, in an ideal scenario all of this energy is converted to kinetic energy, Ek. So;

Egp = Ek

mgh = mv^2/2

You'd then just get the height using trig, as you seem to have done.

HOWEVER,

the task specifies there is friction. This means that only a part of that gravitational potential energy will be converted into kinetic energy.

To factor this in, we need the total energy transformed into friction. Then, we can account for it in the formula as follows (as is explained in the video):

Egp = Ek + Efr

mgh = mv^2 + Efr

As the total energy transformed into friction starts at zero, and we know that work is equal to the change in energy (W=ΔE) and force times distance, we can calculate the energy used up on friction at the end by putting the two things into an equality as they're both equal to the same thing, work (that is, this specific work; just because two things have the same symbol doesn't always mean they're the same thing! symbols are just names we give to values, much like us humans have names).

ΔE = Efr

ΔE = Fd

Efr = Fd

Then, having calculated this, we can figure out the rest. Just solve for v^2.

Now, as for the formula you would've like to use;

You absolutely can use it! The reason the problem was solved in the video differently is that it gives you a better idea of what's actually happening physically. While formulas are handy, you should first understand (or at least try to, some require math way beyond what one would know when they're first introduced) why they work before applying them so you don't later find out that your foundational knowledge is full of gaps. This can be a bit tough if you've been taught physics all your life as just a bunch of formulas that are just because they are and which you have to memorize, but luckily we have the internet nowadays.

The reason you got a wrong result is due to an error in calculation. You forgot to subtract the force of friction from the force you calculated. Unless you're sure in what you're doing, I'd always recommend you draw a free-body diagram.