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If this is a weaker magnet, I wouldn't expect it to cause many problems. Of course, you do need to see if most of it was vomited back up. The problem with stronger magents that are in multiple pieces is that you might have two pieces in various parts of the intestines that might stick together through the intestine walls and lead to damage. If this was a weaker magnet, then that would not be likely to occur.
You will need to monitor him closely for additional vomiting, loss of appetite and inability to defecate. If these occur, you will need to have him seen by your vet. this might indicate an obstruction is present.
Definitely try and determine if all the magnet has been vomited up and if not, check the fecal matter for several days to ensure it is expelled. You might switch to a bland diet to give her stomach a break. Boiled chicken and rice works well for this purpose. I would continue monitoring for about 72 hours to be on the safe side and if she is acting normally, eating and defecating, then she should be ok.
Hope this information is helpful to you.
Dogs do pick up on how you feel. So she may be sticking close to you because you seem worried which worries her. Try the boiled chicken and rice, starting with small portions and see how she does. If she eats it with no problems, then just monitor her for a few days. If she does continue vomiting, take her in to be on the safe side. She'll likely be fine, though if it looks like the majority of the magnet has been expelled already.