What is the amplitude of the impulse resonse I'm learning how to construct a discrete signal as a sum of impulse responses of the system. The final formula is then:

$$
y[n]=  \sum_{k= -\infty}^\infty x[n] h_{k}[n] 
$$

My question is, what is the amplitude of the impulse response, because i thought an impulse response was already a scaled response to the unit impulse inputted into the system? But here they are multiplying again with a scaling factor of x[n]? So is the unit impulse's amplitude just 1 then?
In case of discrete signals, the amplitude of unit impulse is 1. 

It does not mean that your impulse response should have unit amplitude. The impulse response can have any real valued amplitude in it. 

Note: The area under unit impulse in continuous time domain is 1 not the amplitude. The amplitude of a unit impulse approaches infinity as the area under remains constant (=1).