Is it possible to build a jet engine that utilises all excess heat?
No, any thermal machine always transfers energy from the heater to the cooler (Carnot cycle). The maximal yield depends on the temperature difference and cannot achieve exactly 100%. It is equal 

$$
1-\frac{T_\text{C}}{T_\text{H}}
$$

where Tc is the temperature of the surrounding air and Th is the temperature of the burning fuel. Jet engine as normally understood in aviation is a thermal machine.