I have a contact that arranged my permanent residence (it took three years mind you). He told me he has 10 to 15 turnos per month for the coming months.
If you want to get his contact details please send me a private message.
Does your contact have turnos at migraciones (for the residency) or the registro (for the DNI)?
In most cases (when upgrading from temporary), permanent resiency takes three years whether or not you use a contact. Both permanent and temporary residency are granted at migraciones. They do not issue DNI's.
Migraciones is presently granting turnos whether you have a "contact" or not. The regisgro should be accepting turnos for the DNI again in mid to late October. The cost that is still posted on their site is 15 pesos. Getting a DNI is simple compared to getting residency (either temporary or permanent).
For the DNI, all you need is your birth certificate (with Apostille/translated and legalized in Argentina), a certificado de domicilio, and two photos (available inside the registro), and the paper from migraciones stating that you have been granted the temporary resident status.
The latest info I had (in the past week) was that advance apointments were not needed at the registro for the DNI renewal, but there is a limit of 200 per day and you may need to get there and wait in line as early as 4AM.