How do I get my friend to go to therapy?

I (17, male) recently have been made aware that a friend of two years (16, male) has been suffering with severe clinical depression since primary school. He has told me this and several other things:

1: His mother, who cares for him while his father works, is stubborn, stressed, and overbearing. Despite this, he still believes her to be a good parent and wishes her well. However, because of this, he also believes that she would insist on accompanying him to therapy.

2: Due to his upbringing, he has trained himself not to show the true severity of his emotions. This is why I have only recently known the true scope of his struggles. He claims he does this to avoid worrying those around him.

3: He has considered suicide multiple times, including writing a note in 6th grade, for which his mother brought him to therapy. He claims that this incident has caused him to be more covert with his symptoms.

4: He feels as though asking for help makes him a burden to others, no matter the situation or cause. For this reason, he requests that I do not help him, and resists all attempts to bring him to a licensed therapist

5: He has revealed that his older sister, incidentally a therapist by profession, suffers from a similar relationship with his mother, but is more vocal about it. He says that this is already enough stress for his mother

6: He stabbed himself with a pencil hard enough to draw blood a day ago over a few missed questions on an English quiz

Given these circumstances, I offered up several solutions. Since his mother was unlikely to let him go to therapy alone, he cannot ask her to take him there. He claims that his sister is too stressed by her similar situation with his mom to be effective as a therapist, and also that she holds too much of a stake to be an effective therapist. He also says that he cannot go on his own, as his mother is too overbearing to let him out of the house for an undisclosed reason. He also feels horrible about lying in any capacity, so simply telling her that he is going out with friends is not an option, and our school, as far as I know, has no dedicated counselor. Every solution I propose, every time I try to empathize, every time that I try to reassure him, he claims that his entrenched web of interconnected problems negates them all. Seeing this, and seeing that every way I try to help him or have him get help makes him feel like a burden, I reasoned that the only way to overcome this barrier was to make him feel worse about not doing so. So, on impulse, I produced a sharpened pencil and told him that for every day he refused to get help, I would stab myself on the wrist once. To this, he responded by telling me that every time I did so he would do the same. In retrospect, this does not seem like the wisest decision on my part, but the aftereffects are yet to play out.

Before anything too bad happens, what can I do to rectify this situation?