Rod, Gear Shift Link(l:165)

Accurate gear changes start with the front of your left foot, through the control linkage to the change shaft to the selector drum & forks. Any play or wear with the controls will make for a less positive change, wear or damage on the selector forks will further compromise the change quality.
So, a full and careful inspection of all the elements, with replacement of those parts worn or damaged, is the only way to achieve nice sharp, slick gear change that are is part and parcel of enjoyable bike riding.
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Renew all the interfacing parts for the best result.

Order new gaskets and gasket sealant, try this: 08C70K0234M the choice of many trade professionals.

Are there any other parts that may be better being renewed than repaired in the transmission?

This component may have other applications within the Suzuki model range - so please ensure you order the correct part-number for your model.