Yes, all correct.
Batteries are rated in Ah - your battery is 2.5Ah (not 2.5A) meaning it can supply 2.5A for 1 hour or 0.25A for 10 hours or 1.25A for 1/2 an hour. So to sat it contains 9W is wrong (sorry, I started that) and it should be Wh. This is why we can use the number to calculate the time the battery will last.
Calculating your series resistor was also easy. I know that 50mA is one twentieth of an amp so to drop 4V requires 4*20Ω.
After seeing this thread I googled 830nm and found some very interesting articles. This being the most informative,
Background: Aggressive, or even minimally aggressive, aesthetic interventions are almost inevitably followed by such events as discomfort, erythema, edema and hematoma formation which could lengthen patient downtime and represent a major problem to ...
I also checked the price of 830nm LEDs and found
these. Less than ¼ the price and twice the output. This is the Australian Farnell site.
Mike.
Edit, The boost converter you can think of as watts out = watts in * efficiency (normally ~95%).