I accidentally drove with my handbrake on for 2.5km?

I'm a mechanic, and I want you to understand something. The hand brake, parking brake, emergency brake has a purpose, and that is to make your car unable to roll.

Most people think that to use the brake, you just pull the handle up a bit. But that's not true. Pull it up till it gets tight, and then pull it another click or two. Then try to move the car. You will notice that the car may move, but not easily. So if you're not going to put the brake on hard enough to keep the car from moving, then don't bother with it.

Now, back to your worry. If the handle was up just a slight bit, and you didn't notice a nasty burning smell, then everything is probably ok. You could have your rear brakes checked by a mechanic, to make sure. The parking brake mechanically applies the rear brakes.

In times before the mid 1960s, if there was a leak in the brake hydraulics, the brakes could stop working. In those times, the emergency brake, was a pedal to the left of the gas, brake and clutch pedals. Now, it's mostly a lever between the two front bucket seats. In modern times, there are 2 separate brake hydraulic systems, for greater safety.

If you have an automatic transmission, you really don't need a parking brake. It was made for a manual transmission car that is parked on an incline. Because such a car can slowly move. The "park" in an automatic transmission holds the car from moving. This is the equivalent situation. An automatic transmission in "park" and a manual transmission with the parking brake firmly applied. Neither vehicle will roll.

Nothing to worry about

Nothing to worry about. Your car will be fine.

I have done that many times and nothing has gone wrong.

your car will die