Foketry: To begin with Hall-Effects aren't as simple as most people think. They come in different volt & amp ratings, polarities and non-polarity, latch & non-latching, transistor or FET, open collector/drain or open emitter/source, and sensitivity. So as you can see there is a lot of different varieties. You say your KY-003 uses a 3141 Hall-Effect, which is now obsolete and is no longer made and I was not able to get in-depth details on it. However Allegro says it's replacement is the A1104, there is a big difference between the 2 with the biggest being the 3141 is a transistor and the 1104 being a FET. The 3141 has what is known as a open 'collector' and the 1104 having a open 'drain'. In both cases the emitter/source is internally grounded to the GND pin. So when the Hall-Effect turns on it grounds out the signal pin. Now from the info on the KY-003 you posted the circuit shown will not work and if it ever did then it was by fluke. The reason is because the capacitor that goes from the signal to the base of the transistor are both basically at ground level voltage and the cap may only charge up through junction leakage. I simulated the circuit in the PDF and got almost no signal out to the CDI, less than 0.01 volt. I redesigned the circuit so that it outputs positive pulses 11.5 volts (see attachments).
You will need to place a small weak magnet behind the Hall-Effect, the front face is the one with the brand information. You will need to place the south pole against the Hall-Effect. You will need either a compass or a pole detector or do what I do sometimes and use some clear thin tape to hold the magnet against the Hall-Effect. Power up the circuit and then pass a piece of iron pass the front face of the Hall-Effect and see if you get a signal. You may have to adjust the distance between the iron and Hall-Effect to get a signal. If you still don't get one then turn the magnet over and try again. Once you get a signal mark the magnet so you know which way to epoxy the magnet onto the Hall-Effect. Note: use an LED with a 20-40 ma rating.
Ray