First question: is your 60 day visa an extendable one?
If not I think you will have to do the visa run.
If it is then you should be able to get it extended without a visa run.
Sorry this isn’t exactly factual, hopefully someone will shed some true light onthe subject.
If your employer is genuine about the work permit then you should be able to get an extension on your visa from the police station, but you will probably need to show them your work contract, your passport (or something to show that you permits are currently being processed).
If you haven’t signed a contract then your boss is not getting you a work permit as they need to send in a signed contract, photocopies of your passport, etc. to get this rolling.
I switched from a studetn ARC to a Work ARC (Alien Residence Certificate) which is what you will need to remain and work here. However, my ARC expired midway through the process. so we went to the Police station, showed them the documents and they extended my ARC for an extra month so we could complete the process.
This is a little different to your situation.But if everything is above board with your employer, the police should be helpful.
Just a thought. When you went to extend your visa, did you take any documents showing why you are here and that the process of said work permit is in progress?