Does a lens have its own shutter speed, or is shutter speed set by the camera body?

As mentioned in comments, lenses don't have a maximum shutter time. They just let light through continuously, blocked by the camera's shutter from hitting the sensor.

You are also not limited by the camera's internal shutter speeds [at the long end]. Even though the camera's maximum internally-timed shutter is 30 seconds, it also has two 'long' modes, Bulb & Time. These are available in manual mode only.
They are essentially the same thing, but triggered slightly differently.
Bulb will keep the shutter open so long as the release button is held, Time will open on the first press, close on the second.

This would all be made considerably less shaky by using a cheap wired remote* from eBay [$£€ 10 or so]. Also using either live view, mirror up or a shutter release delay.

*These things are very basic, and tend to have a user interface that makes a 1980s digital watch look the height of GUI design, but at most simple you don't even need to put batteries in them. They will work in normal, bulb or time mode simply by pressing/holding the release button. They even have a half squeeze focussing, exactly like the on-camera release.