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Most keyboards simply use two contacts per key, configured so that they close (or open) at different positions in the key's travel. They estimate velocity from the time that elapses between the two events. Even the fancy weighted "piano action" keyboards use this basic sensing method.

Trying to measure position and/or velocity directly sounds like massive overkill.

I have to measure the position of each key in order to know how loud the sound should be.

Normally key velocity is calculated by measuring the time between the normally closed up-switch breaking and the normally open down-switch making.

^{simulate this circuit – Schematic created using CircuitLab}

Figure 1. A break-before-make keyboard contact is typically used for velocity measurement.

The volume of the sound is not only function of the velocity of the keys at the end of the descent. ...
The length of the maximum displacement is about 2 cm.

In this case you need to mechanically arrange the normally closed contact so that it breaks in the last few mm of travel.

^{simulate this circuit}

Figure 2. The mechanical switching arrangement.

It seems to me that your best bet would be to modify an existing MIDI bass pedalboard to suit your purposes.

OP's comment to Dave Tweed:

I think this method is insufficient. Here's why: if I slowly push the key down half of the path, and then quickly push it down to the end, the sound would be loud on a real piano, and this method would think the key was pushed really slowly and therefore output a really quiet sound. In addition, if I want to repeat a note, I would have to let the key completely return to its original position, which is not at all what I would do on a real piano.

That is why I proposed activating the changeover switch in the last few mm of travel.

You should take a look at analog keyboards. Take a look at this video. This is just an example of concept.

Edit:
Also, if optical switches are not working for you, take a look at this:

There is a PCB printed coil under each key cap. As you press the button coil measures the inductance change caused by key spring compressing and decompressing, therefore, you can get pretty accurate reading of the switch position.

Source:
Reddit
Imgur

If the important variable is the impact of a "hammer" that you're trying to emulate, consider piezo sensors which allow you to measure this directly. They produce a pulse whose amplitude depends on the impact impulse.

One option could be Hall sensors, which sense the strength of a magnetic field. There are different types of Hall sensors for digital and analog sensing, you need one that has analog. For example SI7211 costs about 0.80 USD.

You also need a small magnet underneath each key. When the magnet gets closer to the sensor, the magnetic field increases which increases the sensor's output voltage.

You can treat this as a hybrid digital/analog instrument by placing microphones[1] inside the keyboard, tuned to the sound of striking the keys. Signals from the microphones augment the traditional switch-based input, so the switch tells you which key is pressed, and the microphones tell you how loud the most recent keypress was.

2 or more microphones along the keyboard's length would permit decoding a chord with loud/low notes and soft/high notes and vice versa.

[1]: I keep saying "microphone", but I mean "generic vibration sensing device".

You may want to research spring return variable resistors, spring return potentiometers, or linear position sensors. Here is one example:
http://ecatalog.beisensors.com/item/linear-position-sensors/linear-position-sensor-9600-series-compact-spri/9610r3-4kl2-0

Another possibility may be to use a small rotary encoder (and mechanically convert the linear key press motion to rotary motion). The encoder would output one or more sets of pulses as the shaft rotates. A higher pulse rate would indicate a higher velocity press. The encoder position could be tracked directly if the pulse groups are sent to additional digital logic. Here's an example part: https://www.mouser.com/ProductDetail/Bourns/PEC16-4220F-S0024?qs=6FD5PBp7ZtQte%252Bg7b%2FiMUw%3D%3D&gclid=EAIaIQobChMIrKbIjOSa4QIVCEsNCh3JAAKuEAQYAyABEgKM_fD_BwE

If you want super-accuracy at a reasonable price, how about using a linear encoder, attaching a Gray-coded strip (which you print yourself using a laser printer onto a transparent sheet, then cut up and attach one to each key) - more details of how they work under rotary encoders here. This way you need two (perhaps 3 so you get an accurate indication of end position) digital lines per key. This will allow you to measure velocity with very good accuracy and even position if that's relevant. The advantage of this is you could retrofit it to an old keyboard (even an acoustic keyboard).

Caution: when I was a teenager (a long while ago) I wondered for ages whether I could make a realistic keyboard more cheaply than buying them. It seemed unlikely then and it seems even less likely now. The cost effective method is therefore probably "buy a velocity sensitive keyboard and take it apart" which is no fun.