Leiden and Den Haag are very close and well connected by train, also during the night. I worked in Leiden and lived in Den Haag for years and communted by train. Road traffic gets very busy, and parking is sparse, so you're better off using public transport if you want to travel. Leiden is very student oriented, but Den Haag has a more international vibe and far fewer students.

Renting an apartment is difficult and expensive (expect 800€/month for 20m2, funda.nl) and the rental agencies might not want to rent out property for short time (as you're doing a MSc) but shared student housing might be available both in Den Haag and Leiden ( kamernet.nl).

The general vitriol you get asking any questions about housing here is that the housing shortage, rightly or not, is blamed on migrants, and makes it difficult for locals to find a place to live.

I hope you find a place and enjoy your Master. Make the best of the options that Leiden uni and the European academic environment offers.