If it were me as a teacher and this happened, it would go like this. The student doesn't do their work properly, and I lose trust in their ability to do work. However, I wouldn't let that affect my ability to relate on a person to person level with that person. If my time allowed for it, my job would still require that I offer help to students.
Don't be overly apologetic. To me, that shows a weakness that means you're not overly serious about the work. It only means you're concerned about fixing your good name. Be straightforward, sincere and professional. If you're sincere about doing better and show results of the fruits of your labor, that'll help to regain the trust.
Honestly, I don't think you've pissed him off too much. He'd have to be an unhealthy ISTJ to get seriously riled up over it. To ISTJs, work and emotional responses are separated. I'm not sure how other ISTJs handle it, but to me, remaining professional at work is more important than being emotional toward a student. It's more like you're wasting his time.
If you confront him about getting help, I would suggest:
- Be honest about how you're aware that your relationship has been affected recently
- Remain professional
- Be detailed about your desire to do better
- Explain measures you have personally taken to correct your mistakes on your own time
That's the best I can offer right now. For an ISTJ, it's about results and sincerity. The state of the personal relationship is the least of our concern unless it's affecting something tangible.