While i can't do a one arm pull up, i can't imagine it would be that much different to teaching someone who can't do any pull ups how to do one.
1.Do some negatives. So do a two hand pull up on the way up. then hold yourself up at the top with one hand, and on the way down use one hand. Try and resist on the way down. Try to aim for the eccentric phase lasting about 3-5 seconds.
2.Use a resistence band to aid your one hand pull-ups. This will make the movement 'easier' so that you could practice more reps. (https://www.rubberbanditz.com/info-center/pull-up-bands/) link to show how to use.
3.I would suggest not going to failure so much. You've stated that you can only manage three. I would suggest doing no more than two in your regular training. The reason being is that you will be fatiguing yourself. You've mentioned that sometimes you can't even do one. This is most likely the reason (provided your diet and training are consistent). you could start by testing your max only at the end of the month. Although really the less you max out the better.
- you could look for a pull up routine online. There may even be a one arm pull up progression routine out there. Stick with it for a couple of months and see how it goes.