Adding to what Alec has said. You mention feeling same tired after running once a week.
To see the improvement in cardio you need to run at least 45 minutes (not the distance) for at least 3 times a week.
That of course in turn will increase your calorie burned weekly. Which you should cover by eating more.
But remember that just eating more may not be right statement. If you are looking to increase stamina all you are saying you want your muscles to be stronger and have more endurance (that includes your heart which is also a muscle). Saying that - eat more protein rich food - meat, eggs, milk etc.
UPD: Also just running with comfortable speed is not going to increase your stamina and show you good results. To actually see improvements you need to run above comfortable speed. In running, with pushing your limits and having sufficient nutrition (to restore what you burned and have enough material for muscles to recover from stress) you will see improvements in less than a month. But this comes with a big fat WARNING, you need to make sure that you have no heart issues before you try to push your limits.