Why is the kunyomi used in this case?

I don't think there's any particular reason. There are no grammatical rules about when to use on'yomi or when to use kun'yomi - which one to use in which situations needs to be learned through experience.

There are some "rules of thumb" about using the on or kun readings, but they are NOT hard rules by any means. There are many exceptions which do not follow some of these rules (particularly the third one), so do not be surprised if you encounter kanji which are read differently than how you would expect.

The rules of thumb are:

1. Kanji with okurigana are ALWAYS read using the kun'yomi.

2. Single, standalone kanji are usually read using the kun'yomi, but some are read using the on'yomi. Some can actually be read with both, depending on the context.

3. Compound words (with multiple kanji) usually are read using the on'yomi, but there are many exceptions to this rule.

4. Native Japanese names are usually read using the kun'yomi, but some are read using the on'yomi.

5. Places are usually read using the kun'yomi, but again, there are many exceptions which use on'yomi.

6. Foreign words tend to be read using the on'yomi.

7. In compound words, kun readings tend to go with other kun readings, and on readings tend to go with other on readings. There are some exceptions, though, where some words mix kun-on or on-kun readings. These are called 湯桶読み and 重箱読み, respectively.

So for your case:

兄 is a standalone kanji in this case (rule 2), and so it is read with its kun reading, あに.

五日 is a compound word (rule 3), but it uses its kun'yomi reading instead of the on'yomi, so it is one of the many "exceptions".

Do also note that some kanji compounds don't follow the kun'yomi or on'yomi at all, such as 今日 (kyō) or 今朝 (kesa). This phenonemon is called jukujikun (熟字訓) and words under this category should be learned/memorized as exceptions.