You should try talking to your human resources department about long term disability and short term disability options. Many companies offer insurance for when you need to be off longer. Although FMLA may expire, this does not mean that your employer will not hold your position. I was off work for a little over 4 months with my second son. I spoke to my supervisor and kept her updated on my health, surgeries and recovery. You may be able to extend your return date. Since the difference is just a couple weeks, most companies would rather wait for you than hire someone else that has no experience and they will have to spend time and money on training.
Another option is asking if some of the work could be completed at home temporarily. You might be able to find a babysitter, friend or neighbor to help you out a few hours a day while you work. Since infants tend to sleep quite a bit in the beginning, you may be able to get more done than you think. You have to be organized though. Pump milk ahead of time, have meals ready for the day, and wait to wash bottles and dishes until your work schedule is done. I used to carry my little ones in a infant carrier on my chest to do dishes, errands, housework, etc.
Another option is asking if you can complete work on a flexible schedule at home between taking care of baby or when your partner is available to takeover. That might make the time of needing a sitter less.
Additionally, I do not think that you should limit yourself to returning to work just because you cannot enroll your child into daycare. There are multiple other options. If your work simply will not hold your job for another couple weeks, then try to find a nanny to come to your home. There are professional nanny services that can provide you with their certifications, work history and references. Obviously, we are all very careful with our newborns. You have the phone numbers of the prior families, so you can probably begin a rather casual conversation and receive answers to those "sticky questions" that recruiters could never get away with in the workplace. Although the nanny service may be more costly, it would be less costly for the long run if you are able to keep your job.
Also, if it's only for a couple weeks, you may be able to find another provider to watch baby for that time. In my area I call Early Childhood Alliance (EAC). They give referrals to babysitters and licenced providers in the area. They also have a website to obtain referrals. I think it's like a 24-48 hour turn around. They generally call you and then email or mail the information. Licensing agencies for daycare providers can generally direct you to the local service that provides referrals.
You might also check with local elementary schools and see if they have a bulletin board with locals that babysit. There are several schools in my area that allow those items to be posted in a commons area. My sons' school has a list of parents that want to carpool, and I call some of his friends if I need something really quickly.
You might also try contacting lactation consultants and the birthing center at your local hospitals. That is an area where there are a lot of new mothers probably in the same situation... needing a new babysitter in a very short amount of time. Even if you found someone waaaay across town, it would only be for 2 weeks until you could enroll baby in the daycare of your choice.