You are confusing the MRT and BTS
MRT is the underground, and correct, the HI Silom is far from there. It is however within walking distance of BTS (the Skytrain) Surasak, which is what I was referring to.
Lumphini is definitely worth visiting for a walk in a calm oasis in the middle of the city - the noise and all sort of disappears. It is also good for people watching (top tip, walk in the opposite direction to the exercise path
). One loop is around 2.4 km if I remember correctly.
Chinatown is an experience, but it is extremely chaotic on a good day. Not sure you would like it, but for sure, if you feel like exploring you should give it a try.
You could actually also do a day-trip to the old capital city of Ayutthaya - that is pretty interesting and will give you a bit of a break from the busy city. Not sure if it is still an option, but decades ago I did a day trip with a car outbound,
and a boat on the return (the Chao Phraya runs through Ayutthaya and Bangkok). I remember it as a good experience.
If you are debating whether to extend your stay in Singapore vs. the current time in Bangkok, my personal opinion would be a resounding no (both from a places of interest and a cost perspective).