Jotting down the info here, for those who happen to look for it.
Regardless of what's been said on the news regarding changes on KTKLN (Kartu Tenaga Kerja Luar Negeri) system, currently we are still pretty much stuck with the existing system. When leaving Indonesia, an FDW needs to have a valid KTKLN.
A KTKLN can only be issued by BP3TKI or P4TKI office in Indonesia. Complete info can be found here.
One of the requirements is a contract - it has to be issued and stamped by Indonesian embassy (KBRI). FDW and her employer can come to KBRI in the morning (9am onwards) to get a contract drawn up. You'll need to bring along FDW's original passport and work permit.
Additional tips: to get somebody to actually pick up your call in KBRI, your best bet is to call in the morning, say around 9am. If you are not pressed for time you can also email them - I got mine replied after 9 calendar days. I tried posting my questions in the query form in their website, and also tweeted them - both to no avail.