IMHO you are absolutely fine...I would leave the cover off for the initial period, till you get down to say 140 - 150. Sometimes when i brew at night ( 8 gallon batches), I will just leave the kettle chilling in a bath till the next morning and then pitch the yeast. I have a wimpy 25' 3/8 chiller that I rarely even use. I find it easier to just set my kettle in a big 25 gallon tub w/ water, then again w/ water and ice, sort of set and forget. I stir in the beginning and then usually lose interest and do something else, then just pitch when the wort is ready. Oh, having access to a commercial ice machine and tossing in 40 pounds of ice doesn't hurt either.
A big tub, will work faster than the sink if you have that option as it will hold a lot more water. Go ahead and get a chiller when you are ready, but be warned that in the middle of summer, if you have warm tap water, the chiller loses its effectiveness, and you will likely have to use ice either w/ a bath, or prechiller anyways. I prefer to pitch in the low sixties, and unless the tap is running quite cold, a standard IC ain't gonna get there too quick either.
Don't listen to the advice that you absolutely need a chiller...that's bunk...I got one and don't even use it. Using an IC is still work..get it out...put in the kettle...hook it up...stir...wait...stir wait...get frustrated when you stall in the eighties...been there.