He was a _______ writer, producing works that are informative and lively.
The OP's task was to transform the word SKILL in order to fill in the blank space.
But if the OP had to find a word, they could have opted for any of the following: gifted, talented, accomplished, or even polished.
Note that none of the preceding adjectives have the suffix -ful. In fact, giftful, talentful, accomplishedful, and polishful are all ungrammatical today. But in the case of skill, either one of the two suffixes -ed (skilled) and -ful (skillful) can be applied.
Oxford Dictionaries says that skilled means
1. Having or showing the knowledge, ability, or training to perform a certain activity or task well.
1.1 Based on or proceeding from the ability to do something well.
1.2 (of work) requiring special abilities or training.
Whereas for skilful, there is only one definition
Having or showing skill.
- ‘Not only is he a brilliantly skillful footballer, he is also captain and leader of this Czech side.’
- ‘For thousands of years, this has been admired as the most skillful accomplishment in war.’
The writer in the citation can be called skilled because they have the knowledge, and the ability to produce informative and entertaining works. They can also be called skilful because they have a natural talent or have acquired the ability (i.e. the skill) to write well.
In conclusion, both forms are more than acceptable. The answer "skilful", supplied by the OP's book, was unnecessarily restricted.
Google Ngram showing the British Corpus agrees that either adjective is suitable for writer, but skilled writer has a narrow advantage.
Google Ngram with the American English spelling skillful, suggests that skilled writer is also the preferred form since the mid-1980s.