How fast should wort be chilled?

Don't worry.

I've been brewing for over 10 years and I always just let the hot wort cool on the kitchen counter over night. The key is to close the fermenting bucket (and out an airlock in) so that nothing gets in. As a side effect the hot liquid also kills any bacteria that might have gotten in the bucket before.

There are several reasons why wort should be chilled as quickly as possible. One is to get through the "danger zone" between anti-microbially hot and cold as quickly as possible. The other is to promote "cold break" which is the settling out of proteins and other compounds during chilling. The quicker you chill, the better the cold break and, generally, the clearer the beer.

However, "as quickly as possible" is just that. If you don't have plate chillers and refrigerated glycol below freezing point (which is what commercial breweries use) you use whatever you have. A reflux chiller with refrigerated water is better than an immersion chiller with unrefrigerated tap water, but that in turn is still a lot better than letting the wort cool down naturally overnight. Putting your brewing pot into a tub of water orice is also a lot better than nothing.

The rule of thumb is indeed "the quicker the better" but there is no hard-and-fast rule on how quickly the wort MUST be chilled in order not to ruin your beer. So do the best you can, and only if the results are disappointing (microbial infection, chill haze or other defects related to chilling problems) upgrade to something more complicated.

If you can't cool with forced cooling, make sure that your brew kettle is covered adequately.

I started out the same way as you, small brew kettle in the kitchen sink, using ice and water from the faucet. Later on I used a spiral cooler. Now I cover my brew kettle with the lid and a heat proof plastic bag sealed well (when still hot) and let it cool overnight.

Contamination (not infection) is very unlikely if you work clean, and in a clean environment. A clean environment is: not dusty, and no fruit flies in the vicinity. Wash your hands and disinfect the tools that you use and could come in contact with the wort.