This Python script takes the archive you obtain from Twitter and produces a nice static site containing all the tweets, in a form suitable for publishing online.
Install the required packages with pip install -r requirements.txt.
Extract the .zip file you get from Twitter into the same directory as this script.
Run python twitter_archive.py.
The output is in a directory called output. Copy the data/tweets_media folder from your Twitter archive to output/media.
I just made this for my own Twitter archive, and I'm not hugely interested in maintaining a generic tool for everybody. If it's useful to you as it is, then that's lovely!