No. In general the accusative case is used with the word "kosten".
- Dativ (dative case): Ihnen
- Nominativ (nominative case) & Akkusativ (accusative case): Sie
The dative case is only used in special cases with "kosten". See: http://www.spiegel.de/kultur/zwiebelfisch/zwiebelfisch-abc-kostet-mir-kostet-mich-a-309400.html and Wikipedia for the pronouns: http://en.wikipedia.org/wiki/German_pronouns#Personal_pronouns
That would be "Das wird ihr nichts kosten." as you have to declinate the pronoun "sie" (she) according to the dative case to "ihr" (her).
Please note, that "Sie" with a capital "S" means the formal "you" in English. Think of the German informal "du" like "though" in Early Modern English and "Sie" like "you". Today you always use "you" in English, but in German you distinct between addressing people in a formal or informal way by using different personal pronouns (Anredepronomen.)
I suggest you to visit Duolingo on a webbrowser. You get great hints. For example the lesson overview of the skill "Dative Pronouns" teaches you, that the dative of "sie" (she) is "ihr" (her): https://www.duolingo.com/skill/de/Dative-Pronouns You also learn when to use which case (Kasus) in German by reading the tips and hints in the corresponding skills, for example: https://www.duolingo.com/skill/de/Dative-Case
Can people actually tell apart “nicht” and “nichts” while speaking? It doesn’t seem like the voice here stresses “nichts” or makes it clear that it’s not “nicht” in any other way (it’s slow so there’s really no need to, but I’m afraid that nobody talks this slow). Of course it can’t be “nicht” here because “kosten” needs an object (two objects actually... is it something that happens often in German, two parts of the sentence with different logical functions that share the same case?) but I think that some other verbs could be used in a negative form without having to specify an object (“I will not eat” vs “I won’t eat anything”?).