I have a 1999 Chrysler 300M. Simultaneously, the Check Engine light came on, the speedometer stopped working, and the car doesn't shift out of 2nd gear. What's the likely problem?

SOURCE: country speedometer doesn't work and won't shift out of 1st gear

transmission speed sensor needs replacing about 40.oo$ at the dealer sits on the right side of transmission looking from the front of the radiator g s.

SOURCE: 2000 chrysler 300m

The 700 is the limp mode, as I recall. One or both of the sensors are failing, and they are not too expensive to replace. Once replaced, then using the OBD tool, you can delete the fault codes and then be able to shift into all the gears.
I have written about this solution in other blogged questions, and so have others. Under the car, on the Driver side of the transmission are two electric sensors. They lay horizontally on the trans just behind where your foot would go on the accelerator and under the floorboard. The electricl clip sockets pull out with thumb pressure on each socket. DO NOT pull on the wires EVER. Unplug each socket, and replace one at a time. Use a 1" deep well socket and unscrew them from the side of the trans casing. Do one at a time starting with the one on the left, the Input side,; Then the output side.
The reason why I replace them both is that they are cheap- about $20 apiece- and REALLY: do you want to go under there twice for twenty bucks? If so, then just replace the one on the LEFT- the Input sensor. You then MUST RESET THE CODES with the OBD tool for the transmission side of the tool, then GO!

SOURCE: i need help

you need to get the diagnostic codes to determine why its in 2nd gear (limp-in). it can be numerous things..anything from a input or output speed sensor to a serious internal trans problem.

SOURCE: 99 chrysler cirrus lxi won't shift out of 2nd, no speedometer

Had the same problem on my 1999 cirrus lxi. Replaced the output speed sensor on the tranny and topped off the tranny fluid... cleared the check engine light and the cirrus ran great. Caution...the output speed sensor reqires removal of the front left tie and plastic weatherproofing to replace. once you find it's locvation, the replacement is relatively easy.

SOURCE: 99 300m wont shift out of 2nd gear and speedometer

You need a reader that can cancel the trasnmission codes. Look to Advance or one of those parts places. Call to ask them if their ODB tool can cancel Transmission codes first before you go there.