Here are the incorrect parts and their corrections:

1. **Original:** "help me solving"
**Correction:** "help me solve"

2. **Original:** "hardly. She is taking"
**Correction:** "hard. She is taking"

3. **Original:** "weren’t enough intelligent"
**Correction:** "weren’t intelligent enough"

4. **Original:** "than he turned off"
**Correction:** "when he turned off"

5. **Original:** "It's worthless staying up"
**Correction:** "It's worthless to stay up"

6. **Original:** "No sooner he had entered"
**Correction:** "No sooner had he entered"

Each of these changes corrects the grammatical mistakes present in the sentences.