Simple and Cheap method to detect object proximity in the dark

From your explanation and other question, it seems you are connecting your IR LED directly to the ADC input. I don't think this will work too well at a distance, the ADC input will probably have quite a low impedance that will attenuate your signal. Photodiodes have a very large impedance so you need a transimpedance amplifier to convert the current to voltage.
I would use something that is designed for sensing rather than emitting, like a photodiode or your IR phototransistor (if it didin't work then you are probably not using it correctly), and feed this into an opamp, then into the ADC.
In the app note you link to, there are plenty of example circuits, all of which involve a transistor or opamp to amplify/buffer the signal. Try one of these and see how it performs.

I don't think that using the signal of an LDR can do much because the circuit already has some kind of ambient light suppression: it's the high pass filter at capacitor C8.

I agree with MikeJ-UK that the signal probably is saturated by ambient light.

If you just want to get the proximity sensor working with more ambient light I'd suggest to put an IR filter in front of the detector.

If this is too easy (or you also have a lot of ambient IR light, e.g. because the sun is shining at the detector):
You have to solve the problem of the signal being totally jammed by the ambient light.

Lets suppose the photocurrent caused by the signal is some micro amps or less and the ambient light gives you already some 0.1mA there is only a very very small signal voltage at the input voltage divider (D1/R10). The more current (caused by ambient light) is flowing in the voltage divider, the smaller your singal will be.

Just increasing the amplification doesn't help, because the noise will be amplified too and I think you come into regions where signal-to-noise ratio is what you have to take care about.

So instead of having a voltage divider at the detector a better approach would be to utilize a transimpedance amplifier:

Its output voltage is linear to the photo current. So this will give you at least constant signal level, no matter how much ambient light you have (see also this article about this problem by Bob Pease).

Of course this is only true within limits: if your amplifier is jammed, you can't do much.

So the amplification before bandpass filtering must not be too large. But if you make your bandpass filter narrow enough you can do huge amplification afterwards (like in radio receivers).