There are some really good answers already from people who clearly know a lot about programming; Molgrips sums up my view on programming really well, particularly about programmers being born, not made. My only qualification in this is failing to get my 10 year old daughter interested in coding so treat this advice accordingly:
In my opinion you are getting answers from two groups, hardware and software developers. If you and your daughter are interested in hardware the then the an Arduino or Lego is a good option as you are working with a tangible thing.
If you are interested in software stick with the laptop and use a language with an Integrated Debugging Environment (IDE) that way you can visualise the execution of you code and see why it isn’t working. Don’t try to code using notepad and a separate compiler/linker etc.
Take a look at Pencil Code[/url] and book by David Bau its based on his experience teaching children to code.
If you want the try Java with and IDE there is Greenfoot.
The best advice I can give is find a problem you want to solve and work on it together, There is nothing like having/wanting to do something to motivate your learning. Take a look at /r/DailyProgrammer and try to implement Rock Paper Lizard Spock[/url], Hangman or Guess the Number.
Lean together and treat it as pair programming