Learn Polish: Ankieter wnikliwie przygląda się kilku nowym tendencjom. - The pollster is looking closely at several new trends.

The verb przyglądać się (to look at, to observe) always requires the Dative case. It literally implies giving your gaze or attention to something, so the pollster is giving his attention to kilku nowym tendencjom.

I learned that kilka forces the following noun into the Genitive case (like kilka tendencji). Why doesn't that happen here?

That Genitive rule only applies when the numeral is in the Nominative or Accusative case. In the oblique cases (Dative, Instrumental, and Locative), the numeral stops forcing the Genitive and instead agrees with the noun's case. Since the verb requires the Dative, the numeral, adjective, and noun all simply take their Dative plural forms: kilku nowym tendencjom.

What is the Dative form of kilka? I might have expected kilkom to match the noun ending.

The form of kilka used for the Dative and Locative cases is always kilku. It does not take the typical Dative plural noun ending -om. You use kilku whenever the grammar demands Dative, regardless of the noun's gender.

Could I use patrzy na kilka nowych tendencji instead of przygląda się kilku nowym tendencjom?