How would I use this idiom in this sentence- The memory of the incident has forever ........ into my mind?
I fear you are confusing the verb in the idiom to bear [something] in mind, of which verb the past tense is bore (past participle borne, though this site also allows born), with the verb to bore meaning to drill, i.e. to cut a circular hole with a rotating tool.
bear /keep in mind, to remember:
Bear in mind that the newspaper account may be in error. [Dictionary.com]
The image of drilling into a mind suggests the ancient surgical procedure of trepanning, though in idiomatic use to trepan more often suggests the extraction of secrets than forcible indoctrination.
You might alternatively be thinking of a different idiom or metaphor, where particularly vivid and thus memorable impressions are said to be burned into the mind. (I am not sure whether that is a metaphor from wood-burning, as a graphic technique, or from etching. Is Locke’s metaphoric tabula rasa made of wood or copper?) Nowadays this last idiom is often replaced with the observation that “you can’t unsee” something.