Flurry of Blows is the Last Action
Each of the unarmed Strikes you make are considered subordinate actions to the single-action activity Flurry of Blows.
Using an activity is not the same as using any of its subordinate actions. For example, the quickened condition you get from the haste spell lets you spend an extra action each turn to Stride or Strike, but you couldn’t use the extra action for an activity that includes a Stride or Strike. As another example, if you used an action that specified, “If the next action you use is a Strike,” an activity that includes a Strike wouldn’t count, because the next thing you are doing is starting an activity, not using the Strike basic action.
Unfortunately this means that your last action was a Flurry of Blows that damaged the target, rather than the required fangs Strike for Taste Blood.
Taste Blood is probably being overstrict in its requirements here, it's doubtful the game would suffer or become unbalanced if a GM allowed this combo.