What you built is a common collector circuit, and the others already try to persuade you to change that to a common emitter circuit. Common emitter is indeed better for switching, but common collector also works if you keep a couple of things in mind.
While common emitter needs less than a volt to drive the transistor, common collector needs a higher voltage. If the LED's voltage is 2 V you need at least 2.7 V at the base to get the least emitter current. To get 20 mA for the LED you need 20 V extra for R1, and you don't have that, so R1 must be a lower value, like 50 \$\Omega\$. Then 20 mA will drop 1 V across R1, and the base voltage will have to be 3.7 V minimum. Then there will be 0.8 V across R2 and the base current will be 800 \$\mu\$A.
That's not how it works. We would have a calculated base current of 800 \$\mu\$A and a collector (or emitter) current of 20 mA, which would give an \$\mathrm{H_{FE}}\$ of 25. But we don't decide how high \$\mathrm{H_{FE}}\$ is, the transistor does. And that's 280 typical. So our calculation is wrong.
You can leave out R2. Then the base will be at 4.5 V, and the emitter at 3.8 V. With a 2 V drop across the LED we have 1.8 V for R1, and then the current is 36 mA. A bit high, let's increase R1 back to 90 \$\Omega\$ to get our 20 mA back.
But wouldn't there be too much base current without R2? No. To get 20 mA collector current we'll have 71 \$\mu\$A base current, the transistor takes care of that. If the base current would increase because the supply voltage increases then so will the collector current, and hence the voltage drop across R1. The emitter voltage will rise and counteract the increase in base current. A similar automatic regulation occurs when the base current would decrease.
So R1 indirectly takes care of the base current and makes R2 superfluous. But you can't calculate base current as (4.5 V - 0.7 V - 2 V)/R1. The resistance seen from the base is R1 \$\times\$ \$\mathrm{H_{FE}}\$. Why is that? Say you increase base current by 1 \$\mu\$A. Then collector current will increase by 280 \$\mu\$A (\$\mathrm{H_{FE}}\$ = 280), and the voltage drop across R1 will increase by 90 \$\Omega\$ \$\times\$ 280 \$\mu\$A = 25.2 mV. So the resistance seen from the base is 25.2 mV / 1 \$\mu\$A = 25200 \$\Omega\$, or 280 \$\times\$ 90 \$\Omega\$.
And that explains why the LED in your circuit lights so faintly: I = (4.5 V - 0.7 V - 2 V)/(R1 \$\times\$ \$\mathrm{H_{FE}}\$ + R2) = 6 \$\mu\$A! It's a wonder it lights at all.