Welcome to the Bunnings Workshop community. Thank you for joining us and sharing your question about water drainage.
I agree with your assessment that it is not ideal to redirect the flow of water to your house. One possibility is to try and track the current drain pipe that is connected to the downpipe near the house and link to it further down the line away from the house. Your proposal of redirecting the water to the middle of your garden might cause water saturation and create a puddle in the middle of your garden. One other option is to look at the next downpipe further along the area that is linked to an existing drain.
When you find the downpipe that is linked to the drain you can then connect to it with some pipe and a pit at the other end. Having a pit near the pergola area slows the flow of water down and does not create congestion at the other end where you've linked up to.
Let me tag our experienced member @Adam_W for his recommendation.
If you need more options for drainage ideas please let us know.