of Firefox already does all that.
A couple of trick questions for you, do you have access to the source material of the files? And are they originally created in another format before being converted to html?
I ask as I don't know of anyone or any system that created lots of, or large, html files from scratch. They're usually created in a program that does have a grammar and spelling checker built in or converted to html from other documents.
I write a lot and write in ODT format ready to provide a print ready pdf file for the printers (already got over a dozen books in print through small printers) and some are also converted to html for display on free story web sites like storiesonline.net - in these cases I convert the ODT file to html and then clean up the code. If the original docs are not html you can run the checks on them and have clean copies ready for the conversion.
Otherwise just run them through on of the many bulletin board type program that allows you to view and work on them like the reply box of this bulletin board where the browser can run it's checkers over the material.
Do you intend to correct as you go, or what. I ask as many programs like Bluefish are created to check web page code also have built in checkers and can be run over the code the same as the checkers in Word, but they have filters to ignore the html tags when they're put in right - the checker notes faulty tags too.