In general, the best way to split a continuous variable into categories is not to do so, except for graphical display.
You have to distinguish here between the model itself and the way that you display the results. Your question suggests that incorporating your variable directly into the Cox regression as a continuous variable works well. If so, your table of the results of the model should be presented that way, without any categorization. The Cox regression coefficient then represents the log-hazard change per unit of your variable.
For graphical display, it can help the reader to display survival curves for subgroups based on the value of your variable. Displaying survival curves for 3 to 5 groups of approximately equal size from lowest to highest values of your variable can be useful. But do not use those subgroups for building or testing your model or for calculating group-based hazard ratios. If a cutoff is, say, at 25 units, the success of your variable as a continuous measure means that there should be little difference between cases with 24.9 and 25.1 units, but categorization with a cutoff of 25 would implicitly suggest a large difference in outcome.
Graphical display by subgroup can help identify limits to a linear contribution from a continuous variable; see this answer for further details.