A pergola won’t help. You will prob need planning, building and body corporate permissions to build it in and very expensive . It needs to have the tiles removed and rewaterproofed, paying particular attention where the door is as that is prob the main issue. Sadly a common problem.

is this a private home or part of a block of units or similar and what's below the balcony?? if it's a downstairs room that's being affected by leaks then i would fully enclose the balcony, repair all the damage and use the new room as an addition to your master suite... if there's nothing under the balcony why not remove it completely...either way there is no cheap option and issues are likely caused by poor construction or lack of maintenance but there are tile and concrete sealers that can be applied to the surface that can temporarily control the problem until you decide on a permanent solution