Kirchhoff's voltage and current law
Why is that Kirchhoff's voltage and current law always give answer?


They don't always give answer.  For example:







simulate this circuit &ndash; Schematic created using CircuitLab

KVL applied to the above gives:

$$1V = 2V$$

which is a contradiction.

Another example:







simulate this circuit

KCL applied to the above gives:

$$1A = -2A$$

which is a contradiction.

Another example:







simulate this circuit

KCL applied to the above gives

$$\frac{V}{1 \Omega} =  \frac{V}{1 \Omega}$$

so the voltage and current are undetermined.

The fact is, it is possible to draw an ideal circuit schematic that is inconsistent or has undetermined voltages or currents.

Furthermore, node voltage analysis relies on KCL while mesh current analysis relies on KVL so I don't quite understand the 2nd part of your question.