Hello Luv!
You are staying in Hell's Kitchen and all you are worried about is buying laundry detergents? *LOL*
There is a huge supermarket (was Food Emporium but by now should be Key Food) at 452 W 43rd St (btw 9th & 10th Avenues).
You can also try Duane Reade at 43rd Street and 9th Avenue.
If you *really * want to put your skates on go down to the East Village and roam about Kmart on Astor Place:
http://www.yelp.com/biz/k-mart-new-york
Detergent/laundry/cleaning supplies are on the lower level (same as subway) and is quite vast and inclusive. Far more stock than you'll find at most Manhattan supermarkets and better prices as well. While you are there might as well take in the whole place for inexpensive things you might want for yourself or gifts. When done (or before) you can either explore the East or West Village; or do both.
From where you are staying the easiest way to get down to the West Village is walk one block west to Seventh Avenue, then few blocks south to 42nd Street and take the "1" (One) local train to Christopher Street. Getting off there walk east along Christopher towards Sixth Avenue. 8th Street is a kitty corner right as you cross the avenue. Then keep walking east along 8th Street until you get to Astor Place and Kmart. It is one block east of Broadway on the North side of the street across from a Starbucks. It isn't that long of a walk and you'll probably enjoy much of it. If you take this route you will pass the *historic" Stonewall bar and see some sights of the West Village. It isn't what it once was as like much else in Manhattan high rents have chased many away but still fun.
There is really much to see and do in the area from 14th Street/Union Square down to 8th street. Truth to tell really beyond but don't want to get you lost. However it is a fun place to wonder about. If you have your phone with any good GPS/map you'll be fine.
In general supermarkets here don't have the best prices for laundry products unless they are on sale. Kmart is more of a discount place so their prices often are better. Duane Reade, Rite Aid, Walgreen's and a few others are sort of hybrid chemists and grocery stores. Sometimes their prices are lower than supermarkets, others not.
If you can find a Rite Aide store they most always have Tide detergent on sale at least every other week. It may be only the liquids though which would cause problems getting back through customs. The pods I don't know if would pass muster either.
But secretly I'm more excited about visiting the detergent section (kid in a sweet shop comes to mind) but I won't let on to him I'm less excited than what he is 
. I'd really like some wisk liquid so fingers crossed I can find some on my travels. 
