How to address today and the following 9 days
Solution 1:
You actually already expressed it—twice—in your question.
Consider the singular case:
I'll do it the next day.
It's a relative term. It doesn't mean today—or the specific day previously referenced in a discussion. It means the day following.
Similarly, the next X days means the plural number of days following.
If you wake up in the morning and want to express something that will happen over a 10-day period, including today, you would say:
I'll be working on it today and the next 9 days.
There is no simpler phrasing that I can think of.
Note that there might be some ambiguity over the use of the word next.
If, at 6:00 p.m., you say the next 10 days, it's likely that you mean tomorrow and the next 9 days. Most people wouldn't bother specifying tomorrow in that case.
But if there's room for misinterpretation, it's always good to be explicit.