Any dynamic mic with an impedance between 50 and 1,000 ohms
should be compatible with any cassette recorder.
(Most low-impedance mics are between 100 and 600 ohms.)
If you use a mic with an XLR connector,
you will have to make sure the cable is wired correctly in order for it to work, but that isn't terribly difficult.
Fortunately, getting it wrong can't hurt anything --- it just won't work.
Electret mics with internal battery power will also work,
but some may slightly overdrive the mic preamp,
which will cause distortion.
Since you will have no easy way to control the signal level,
a dynamic mic is the best choice.
I have used a Shure 545 with a number of cassette recorders.
The results have been as good as the recorders themselves have allowed.
Now, before you go buying or even borrowing a separate mic,
you should know that it is a Rare Occurrence that the internal mic in such a recorder will fail.
It is Much More Likely that either the head is so dirty that it can't make a recording
or the bias oscillator has failed due to a bad capacitor or transistor.
Have a local expert (such as a technician at a repair shop) examine the recorder and see if it needs service.