How to calculate the specific heat of ethanol?
Each gram of water has an specific heat of 4 J/(K.g)*. So heating $\pu{100 g}$ of water from $\pu{15 °C}$ to $\pu{70 °C}$ would take:

Heat = (70°C - 15°C) * 100g * 4 J/(K.g) = 22,000 joules = 22 kJ


$\pu{20 g}$ of ethanol would be:

Ethanol mol = (20g / 46,07 g/mol) = 0.43 mol


So this means that:

0.43 mol = 22 kJ
     mol = 22 / 0.43
     mol = 50677 J = 50.67 kJ


So ethanol has 50.67 kJ/mol vs 53.72 kJ/mol said by you. I would say that you have made fine your homework, there is a very little difference (I have used some decimal to calculate, maybe you not. (Below you can find a more accurate answer)).

* Water specific heat:  water changes it's specific heat when it changes it's heat. For example: (Wikipedia)

Water at  100 °C (steam)  2.080  J/(K.g)
Water at   25 °C          4.1813 J/(K.g)
Water at  100 °C          4.1813 J/(K.g)
Water at  −10 °C (ice)    2.05   J/(K.g)

Normaly it's used:        4.1813 J/(K.g)


A better table can be found in a french page.

Using this table we can do a better aproximation:

Water heat = (70°C - 15°C)        * 100g
Water heat = (293.03 J - 63.04 J) * 100g
Water heat =    229.99 J          * 100g
Water heat = 22,999 J = 22.999 kJ ≈ 22.9 kJ

Ethanol mol = (20g / 46.07 g/mol) = 0.43 mol
Exactly: 0,4341219882787063164749294551769 mol

0.43 mol = 22999 J
     mol = 22999 / 0.43
     mol = 52978.19 J = 52.97819 kJ ≈ 52.97 kJ
Exactly: 52978,196500000000000000000000001 J

Answer: Ethanol has 52.97 kJ/mol. A value closer to your own value.



PD: I am only a normal person who like chemistry, maybe I am wrong. For example in the theory I have no idea what means $Q = c \cdot m \cdot \Delta{}T = \pu{4.2 \frac{J}{g \cdot K}} \cdot \pu{100 g} \cdot \pu{55 K} = \pu{23100 J}$ Well, I have some ideas but I haven't learned it in the school.


Edit:

I notice that you have your own mol of ethanol (46 g/mol instead of my 46.07 g/mol) and your own specific heat of water (4.2 J/(K.g)). You have to do:

Water heat = (70°C - 15°C) * 100g * 4.2 J/(K.g)
Water heat =      55°C     * 100g * 4.2 J/(K.g)
Water heat = 
Water heat = 23100 J = 23.1 kJ

Ethanol mol = (20g / 46 g/mol) = 0.43 mol
Exactly: 0.43478260869565217391304347826087 mol

0.43 mol = 23100 J
     mol = 23100 / 0.43
Using mol decimals (0.43478260869565217391304347826087)
     mol = 53130 k = 53.13 kJ
Using only 2 decimals (0.43)
     mol = 53720,93 J ≈ 53.72 kJ
     Exactly: 53720,930232558139534883720930233 J

Answer: Ethanol has 53.13 kJ/mol (With decimals) or 53.72 kJ (with 2 decimal). So yes! You have done right!