Have you downloaded the Service Manual from this forum? You can find it in the TSB and How to Section, click this link: Documentation service manual
Look at the section for transmission, find code P0715. There you will find a long list of testing. This testing will take a lot of time using an electronic multi-meter as you will be testing all the wires from the transmission to the TCU and then the TCU to the PTCM. There are two wires that are CAN BUS running from the TCU to the PTCM and they are all shown in the service manual.

You can also use the search box in the upper right corner of the forum, click "threads" and choose the google option.Type in P0715 and many threads will appear for you to read.

Are you certain you are not stuck in Limp mode? Does the window in the instrument panel for the P, R, D, all show correctly?

The one thing that really bothers me is how dirty the fluid looked in the picture and you have said that it "slips" and that is why I think it basically comes to: If D0715 is the only Diagnostic code you are getting and all the wiring looks good after testing with multi meter then the only option you have is to decide between replacing the conductor plate (just parts, not including any labor, here in the States with fluid about $260 or finding a good used Nag 1 or Mercedes 722.6 transmission, about $700 from a salvaged car)

It isn't difficult to replace the conductor plate, you inserted the filter into it when you changed the fluid.

With a transmission swap you need to get the car up in the air so there is room to drop the transmission down and out of the car. If you swap transmissions you should replace the rear main seal AND the plate it sets in. Of course you can also pull the engine and transmission up and out from the front, but that is more work, more time consuming and expensive as all fluids must be replaced. Spend the time looking trough the service manual and reading the search threads before making that decision. GOOD LUCK!