The broken files might not be the reason always for sensitivity because it depends on which stage the file has broken because if it is broken after cleaning and shaping of the canal then the broken file (needle what you ref) will act as good seal just like gutta purcha because before they used to use silver cone  which is metal for final filling so that won't be reason always if it is giving seal apically that is at the end of root And if file is broken at middle or at the coronal part then bypass has to be done to clean up the remaining root if you want to save the tooth

As because of that separates file inside your root there are some remaining vital part of nerve causing sensation and pain so better to post the X-ray and go through the rest rct as follows *  retrieve the broken/seprated file * bypass the separated file If both are option not possible then extraction and replacement of missing tooth

The doctor can bypass the broken needle and retreat the tooth. In some cases it is possible to also remove the broken instrument. If the doctor is not able to do either of the above two options then we need to remove the tooth and replace it .

It can be retreated if the infection is nt severe and first we need to see the xray

Visit an endodontist. Based on your xray he will decide if the tooth can still be treated with a re rct and saved or not.