I have an old pair of Wharfedale speakers which stopped playing any music whatsoever. One is still a bit beeping, while other one is completely quite. One was already repaired few years ago, only capacitors were replaced and that seemed to work. Here is an image of this repaired circuit:



Where should I start? What do you recommend?

 

That is a horrible repair job on those capacitors in a speaker cross-over. For proper performance, the caps must be non-polar. Get the proper caps.

How do you know that the cross-over is the problem and not the amplifier or the speaker?

Disconnect the woofer (low frequency speaker) and connect directly to the amplifier and test to see if it works.

 

Here is a picture of original cross-over filter. What capacitors and resistor should I get for replacement?

 

It says it right there on the capacitors:

25μF 50V NON-POLAR
4.7μF 50V NON-POLAR

You do not have to match the voltage rating exactly. It is ok to use higher voltage rating without causing harm.

 

The resistor is labeled, too. something...5%
but resistors are much more reliable than capacitors.
Read the label. Measure it with an ohm meter. If it measures good, ignore it.