Hi NVazquez
I am not sure if the problem is associated with your debug probe. You said the probe is cheap, we can't know if it functions well. I suggest you checking vendor for it, if any driver needed. It's better to test the probe with another board, make sure the probe is in good condition.
We can't provide you suggestion if we don't have exact debug probe as yours.
If view of MCU side, you need to make sure SWD interface design is correct, please refer this
https://community.nxp.com/t5/LPCXpresso-IDE-FAQs/Design-Considerations-for-Debug/m-p/469565#M44
If you follow above design and power is provided on board. there is no problem on MCU side.
Thanks,
Jun Zhang