You're being scammed. Unfortunately, eBay sides most of the time with the scammers. Tell her that her 'repair guy' is either made up or incomptetent because you don't 'change the oil' in these machines. Offer her $20 to replace the 'broken needle'[, or she can ship the machine back to you at her expense and you will give her a full refund once you have it back, as long as it is in the same condition you shipped it in, minus the broken needle. You will have to eat the cost of that.
If she refuses and opens a claim with eBay, tell eBay what you offered her. Tell eBay that the machine was in perfect working condition when it was shipped, and that you are willing to accept responsibility for a needle that could have been broken during handling. But the rest of what the buyer is claiming is an obvious shakedown tactic. Tell eBay you are happy to give a full refund if she ships it back to you and when it arrives and all that is wrong with it is the needle, you'll issue the refund.