Do speakers affect LED monitors?

No they do not. Classic CRT monitors fired electron beams onto phosphor dots on the back of the glass which made them glow to produce the light which you can see. The strength of the beam defined the brightness and chemicals in the phosphor set the colour.

However the beam is just a fine point and on its own would just produce a single dot on the screen. The beam scans across the screen turning on and off as required because of two magnetic coils on either side of the screen and the scan moves vertically because of another pair of magnetic coils top and bottom.

An unshielded magnet close to a CRT screen would add its field to the scan coils of the screen and distort the beams leading to the beam missing its target and corrupt the display. If left in position the field would eventually magnetise the chassis and leave a permanent distortion. All voice coil speakers work basically the same way and this is why speakers sold for computer use were always described as "shielded"

LCD screens do not use any magnetic principles to create the image so do not suffer any effects from an external field. Older LCDs use cold cathode lamps to provide the backlight and in theory a very strong magnetic field can distort the plasma in the tube so might cause some shadowing while the field is present but residual magnetism is very small in comparison so there would be no permanent effects. LED backlights don't even have this problem. A strong field that is oscillating very fast may cause a problem but this would not apply to a magnet.

One mistake often made is the assumption that flat panels do not suffer "burn in" often seen on CRTs. This is not the case, if the same image is displayed for a long time it will partially permanently set the crystals  and produce a shadow on the screen. It does happen so quickly on LCD screens but it can happen. Plasma displays (large flat panel TVs) have a sensitivity to burn in somewhere between the two.