Problem creating an RS Latch using NOR gates.

I have been trying to follow this tutorial (NOR Gate S-R Latch | Digital Integrated Circuits | Electronics Textbook) using the exact same components except for the DIP switches, which I have changed to push buttons because I don’t have any. The problem is that it doesn’t work like it is supposed to: when I press the button the LED lights up, but when I let go it turns off. The other botton does exactly the same thing with the other output. I have been trying to figure this out in my own for about three days now (looking for schematics in the internet and similar problems in the forums) and I just find the same schematic that this tutorial. I’m using a constant 5V power supply from an old computer, and I have checked that the polarities are correct and that the 5V are where they are supposed to be.
If any photos of the circuit are needed, I would gladly post as many as required.

Thanks for any help.
PS: My english isn’t the best, I’m currently learning it and thus there might some errors. I apologize in advance.

Check the voltages coming from the switches are changing as you expect and getting to the actual
pins on the device, check the chip actually has power to its Vcc pin and ensure it has a decoupling
capacitor (that circuit you linked is wrong in omitting decoupling, decoupling should never be
omitted for a high speed logic circuit like this).

Until you've added proper decoupling any logic circuit can be expected to misbehave in mysterious ways.

If you are using CD4001 cmos parts as in the article remember these are very static sensitive and
you must handle them accordingly. Modern CMOS parts probably have better protection but all
CMOS parts can be instantly ruined by static electricity discharge.

I have added a 100 pF decoupling capacitor, checked the Vcc voltage and the logic levels at the pins when the buttons were pressed. They all were normal. I also checked every gate of the IC and it worked without a problem following his truth table. I don't really know what else can I do.

100pF is 1000 times too small. 100nF is typically used.

Oh hang on, that circuit says 470 ohm resistors for the LEDs - change those to 4k7 and the circuit
might have a chance of working - the CD4001 can only output very small currents of about 1mA

Altogether that circuit is pretty awful.

Use 5V. Use a 74HC02 chip. Much better output current, and a modern device, not something from
the 1970's...

I changed the capacitance and the circuit behaves the same way. The Resistance trick that you mentioned worked!!!

Yes, I'd forgotten how weedy 4000-series logic outputs were - the load was preventing the output
voltage to reach a level high enough to count as logic HIGH.

This is a case where all-about-circuits is as crap as an instructible, alas.

There are 2 gates left over - use them to buffer the LEDs.

PS: My english isn't the best, I'm currently learning it and thus there might some errors. I apologize in advance.

You're doing fine. In fact, your spelling is well beyond many English-as-first-language types who don't give a s**t.