Hsk2 :: 300 words
Hsk5 :: 2500 words
Difference :: 2200 words → 10 months → 220 words per month
Totally doable! Learning 8 words per day in ten months is possible, if you use a Space Repetition System (SRS) Method.
I would recommend:
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Download the HSK word lists for levels 3, 4 and 5
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Install Skritter and pay for it (one of the few apps worth paying, for this very specific purpose) and take their course on HSK3, then HSK4 and then HSK4 vocabulary lists, 8 words per day. their SRS method will make sure you practice enough.
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Regardless of the above, print those HSK lists in a physical sheet of paper. Then grab a blank sheet and a pen and practice to write the words by hand. Memory is more properly stimulated this way. Drawing the character with the finger on the phone’s surface with Skritter is not enough, extra practice by hand is required.
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Entering the list in an additional SRS flashcard method like Quizlet may further help, since the answer there may require you to actually type in your computer the words with your keyboard
So you would have a double SRS system (Skritter and Quizlet)
And an extra handwriting practice for better fixation of the knowledge in your brain.
Chinese grammar is not that hard, if you have a strong foundation in characters and words up to HSK5, then you can afterwards learn in a more organic direction after that (I would not recommend this for HSK6 so much)
Also, for passive input at all levels, we have Lingq