I have designed a resume for a client and want to know how I can give this to her to be able to edit it freely and update her resume. I have tried exporting the layout and bringing in the design as a PNG background in Word. But word is stupid! Maybe there is another software that can do this?
She could use InCopy to edit the content, but you will need to set it up so that she can check out the content, edit it and check in back in. You will still be involved. If she doesn't have a CC subscription, it's about US$5 per month.
Otherwise, she will need InDesign to edit an InDesign file.
The best way would be to save a copy of the InDesign file and keep it handy to change it as your client needs it.
The following link is a really good primer on getting started with generating quality résumés with InDesign. Even if you don't like the style of résumé used for the tutorial, the lessons learned will help you create a killer résumé for your client.