There is not a single way to grab a broken needle. Sounds like you did a great job. Now depending on your machine and "if" a needle were to get deep into your machine, a lot will depend on the make and age of your machine. My FW and many of my older machines, the needle will drop all the way through and onto a table or into the cabinet of the machine. On newer machines, tipping it upside down and using a magnet may work. IF your machine is interpreted and not working it may need to be serviced. Much will depend on how handy you are at doing a bit of maintenance on opening your machine to look for the culprit. I know the fear and it takes your breath away.