car out of order

It was sunny and warm today.

Since you are recounting the day's events, I think this sounds better in the past tense.

So I planned to go out today.

It's unnecessary to say "today" again. When you start this sentence with "so" the reader will realise this is a result of the previous sentence. You can also join these together with a comma:
It was sunny and warm today, so I planned to go out.

On such a brilliantly sunny morning, I found my car was out of order!!!

It was because the battery had run out.

Alternatively:
It was because the battery had gone flat.

So I asked for a help from an automobile service to charge the battery.

It's more natural to say that you "called" a service. People will understand this as you calling the service by telephone.
So I called an automobile service to come and charge the battery.

I wanted to go out for a leisure, but I was strongly advised to replace the battery.

"for a leisure" does not sound natural. You can leave it out altogether and the sentence is fine, or you can say something like this:
I wanted to go out for a drive, but I was strongly advised to replace the battery.
I wanted to go out and enjoy the sun, but I was strongly advised to replace the battery.
I wanted to go out and take advantage of the good weather, but I was strongly advised to replace the battery.

So I went to a car dealer to have my car checked and the battery replaced.

I am thankful for small mercies.

Since I had nothing in particular to do today, I wasn't in a hurry.