The correct option is
B broke
The first sentence states a present situation 'I am late'.
The next sentence gives the reason behind being late.
The reason has already happened, hence the current situation 'I am late'.
('I'm' is a contraction of 'I am.)
Hence the sentence will be in past tense.
Option B: 'Broke' is a simple past tense verb and states an action already happened. Hence option B is correct.
Option A: 'Break' is the base verb and does not state what tense it is.
Option C: 'Has broken' is in present perfect tense and states that an action that has just happened. But, the action in the sentence has already happened.
Option D: 'Will break' is in future tense and states an action yet to happen.
The action in the sentence has already happened.
Hence options A, C and D are incorrect.