Normal spoon? I really want to see what these abnormal spoons looks like now... :p
As most feminine nouns, "łyżka" (a spoon) has the following three cases identical: plural Nominative, plural Accusative, singular Genitive.
"mieć" takes Accusative. When it's negated, it takes Genitive instead (the only case that changes when negated). So as now you know that the noun is in Genitive here, it must be singular.
You have to understand what case is used in the sentence. Yeah "łyżki" is Nominative/Accusative of "spoons" (plural), but it's also Genitive of "spoon" (singular). Those three forms are identical for most feminine nouns. So, "mieć" (to have) takes Accusative. But it's negated here, and negated Accusative = Genitive (other cases stay the same when negated!). So if it's Genitive, then it means that it's a singular spoon.
Also you can see it by 'normalnej', which definitely isn't plural.