What is the best platform or tool to use if you want to teach a 10 years old girl basic programming skills and techniques in a fun, logic and inspiring way?

First of all -- does she really want to learn how to program? If so, you can try a general purpose programming language, like Python; or programming environments designed for children, like Squeak or Scratch. Try to find an interesting project that you can build together. Let your enthusiasm inspire her, but don't be pushy. Concepts that seem obvious for us may be too abstract for her. Above all, respect her interests -- don't let your good intentions become a source of doubt and uncertainty for her.

Here are some tips.  They are classroom-oriented, but you may find some helpful info as well:   Good luck!