I've had my 05' Liberty Limited since new...
and have had a problem about once every few months (but twice this past week -- hope it's not getting worse) like this:

Car starts, runs and drives fine, but several things don't work:
The miles per gallon display shows "--- MPG"
The other caluclations requiring speed or distance don't work (trip miles, etc.)
The odometer below the speedo still works
The steering wheel radio controls do NOT work
At night, with headlights on, the displays do NOT dim to the nighttime brightness level, but stay bright -- although the backlight for the guages does come on.

If I turn off and restart the KJ, everything works fine.

It's never been something that the dealer has been able to find becuase of that... has anyone else tackled this problem?

If your speedo does not work that is an indication of a bad wheel speed sensor which is located on the top of your rear differential. I've had to replace mine 3 times due to various off-roading 'incidents'. If your wheel speed sensor is not working then your mileage will not change and your speedometer will show zero. I'm not sure what is causing the other issues though. It could also be that you have a loose connection or bad relay that ties these circuits together somehow.

After looking in the service manual for curiosity sake i noticed that the BCM controls those items you describe. However, that does not necessarily mean the BCM is at fault but it is a common denominator. The BCM has a number of circuits that connect to it that involve relays that could also be faulty. You can get the BCM checked with a scan tool from the dealer.

From Service Manual -
There are two different versions of the BCM: base
and premium. The base BCM is a subset of the components
in the premium version. Basically, the base
version BCM does not support the following features:
Compass Mini-Trip Computer (CMTC), fog lamps
(front and/or rear), Remote Keyless Entry (RKE),
remote radio switches, or Vehicle Theft Security System
(VTSS). Both versions of the BCM utilize integrated
circuitry and information carried on the
Programmable Communications Interface (PCI) data
bus network along with many hard wired inputs to
monitor many sensor and switch inputs throughout
the vehicle. In response to those inputs, the internal
circuitry and programming of the BCM allow it to
control and integrate many electronic functions and
features of the vehicle through both hard wired outputs
and the transmission of electronic message outputs
to other electronic modules in the vehicle over
the PCI data bus.

BCM INFO
The microprocessor-based Body Control Module
(BCM) monitors many hard wired switch and sensor
inputs as well as those resources it shares with other
electronic modules in the vehicle through its communication
over the Programmable Communications
Interface (PCI) data bus network. The internal programming
and all of these inputs allow the BCM
microprocessor to determine the tasks it needs to
perform and their priorities, as well as both the standard
and optional features that it should provide.

The BCM programming then performs those tasks
and provides those features through both PCI data
bus communication with other electronic modules
and through hard wired outputs through a number of
driver circuits, relays, and actuators. These outputs
allow the BCM the ability to control numerous accessory
systems in the vehicle.

The BCM operates on battery current received
through a fuse in the Junction Block (JB) on a nonswitched
fused B(+) circuit, through another fuse in
the JB on a fused ignition switch output (run-start)
circuit, and through a third fuse in the JB on a fused
ignition switch output (run-acc) circuit. This arrangement
allows the BCM to provide some features
regardless of the ignition switch position, while other
features will operate only with the ignition switch in
the On, Start, and/or Accessory positions. All of the
battery current circuits are connected to the BCM
through the JB/BCM connector. The BCM receives
ground through five separate circuits. Three of these
circuits are connected to the BCM through a connector
and take out of the instrument panel wire harness
on three separate ground circuits, while the
other two circuits are connected to the BCM through
the JB/BCM connector. All of these circuits are
grounded through a splice block located in the instrument
panel wire harness with an eyelet terminal connector
that is secured by a nut to a ground stud on
the driver side instrument panel end bracket near
the JB.

The BCM monitors its own internal circuitry as
well as many of its input and output circuits, and
will store a Diagnostic Trouble Code (DTC) in electronic
memory for any failure it detects. These DTCs
can be retrieved and diagnosed using a DRBIIIt scan
tool. Refer to the appropriate diagnostic information.

BCM Diagnostics
The hard wired inputs to and outputs from the
Body Control Module (BCM), as well as other hard
wired circuits for this module may be diagnosed and
tested using conventional diagnostic tools and procedures.
However, conventional diagnostic methods
may not prove conclusive in the diagnosis of the
BCM, the Programmable Communications Interface
(PCI) data bus network, or the electronic messages
received and transmitted by the BCM over the PCI
data bus. The most reliable, efficient, and accurate
means to diagnose the BCM and the PCI data bus
network inputs to and outputs from this module
requires the use of a DRBIIIt scan tool.