Is there way to get to the other layer
Nope. Multi-layered PCB's are resin-epoxied under heat and pressure.
Test resistance / capacitance across visible ground and VCC.
Measure between common ground plane (ADP_GND1 /ADP_GND2 / PANEL_GND) and ADP_VCC1.
No reading means a missing track is buried.
Some reading means maybe, possibly, hopefully the + and - are available.
Thankfully, your photos are very sharp and clear, EGGcelent photography !!
On to the pin-out and polarities:
ADP_VCC1 = positive DC voltage.
GND ((ADP_GND1 / ADP_GND2 / PANEL_GND) = negative.
ADP_GND1 and PANEL_GND are the switched-negative rail of the original socket.
However, looking closely, you can see that these are both connected commonly to the main ground plane.
(The pads all use a narrow trace to bridge to their destination - the socket's oval pads each use 4 traces).
Polarity is found by following ADP_VCC1 via the 3 amp fuse 'F801' then you'll see the 220u 25V electrolytic cap on right has negative stripe and white screen print artwork on opposite side (blue line), making fuse side poz.
Do you know the voltage, or already have an adapter?
Possibly 12V DC centre positive rated 3A or so.
Otherwise research specs on web.
Regards.
Clive.