It can certainly be fixed. If you want to fix it yourself, however, you will have to determine exactly what the problem is.
Do you still have the needle you took out of the machine? I would compare that to a new needle to figure out if part of of the needle broke off in the shaft. It should not have required so much pulling to get the needle out, although maybe it was rusted in?
I would also tilt the machine on its side and take a small, strong flashlight to that area to see if there is rust, if part of the old needle is in there, if the screw has broken off and is stuck in there, etc. It may simply be that there is hardened oil in that area that needs to be cleaned out. What you can do is stuff part of a cotton ball in there and oil the cotton ball with sewing machine oil. Letting the part sit in contact with sewing machine oil for 24 hours or more can help loosen gunk, and then you may be able to use a dental pick (inexpensive at the drugstore) and/or cotton swabs to clean out the entire area.
If the screw has actually broken off, you can probably order a new screw (and maybe the entire needle assembly) from a Singer sewing machine dealership. I usually like ordering parts from this place online:
http://www.sewingpartsonline.com/sin...odel-list.aspx
but I do not see your sewing machine model listed. The online place tends to be faster and cheaper than dealerships. It would be worth an email or call to them to find out if they have the part(s) you need.