The doctors must determine the cause of your mother's pneumonia. Now that she has a tracheostomy, it would be easy to obtain mucus specimens and even to biopsy a sample of lung tissue for further testing. You should ask her doctors if they will obtain specimens in this fashion. Also ask her doctors if she could have severe acute respiratory syndrome (SARS).
There are new techniques to identify infectious organisms but conventional cultures would suffice. Antibiotics continue to evolve to address infections that were: 1) previously resistant to our current antibiotics; or 2) improperly cultured.
I am a registered medical technologist and have worked in microbiology for over 30 years. The third drug listed is Levofloxacin. The first two I'm not sure of, as you may have the spelling wrong and/or the name they're referred to in Hong Kong may be different from the one here in the US. If you can get a better spelling, I may be able to help you with the names.
If she is in ICU, I'm sure she has been cultured quite a bit. Levofloxacin is a member of the same family as Ciprofloxacin, which is the Quinolone family. These are broad spectrum antibiotics, i.e. they will hit a number of different bacteria. When cultures are collected, they are sent to the laboratory and plated on Petri dishes (different types depending upon the site) and incubated overnight in an incubator. They are examined for pathogenic (disease causing) bacteria on day 1 and day 2 of incubation. Depending on what is growing, an antibiotic susceptibility test is set up on certain pathogens. Other pathogens get identified but they have known patterns of antibiotic response and don't get AST's. If fungal infections are suspected, fungal culture must be requested, as incubation for these specimens is at least 4 weeks instead of 2 days.
Your mother is at risk for resistant bacteria that flourish in ICU settings, especially with the introduction of a tracheal tube. It may not matter at this point what originally put her into ICU. What matters now is keeping her from becoming colonized and/or infected with resistant ICU/hospital bacteria. The biggest risk to her is MRSA (methicillin resistant Staph. aureus) and Acinetobacter baumanii (a very resistant bug that rears its ugly head in some ICU units). Is there any way you can insist on an infectious disease doctor taking over her care? Getting her an ID consult can be a very important thing. If she is already getting ID consultation, she is probably in pretty critical condition and they are trying whatever they can to eradicate the pathogen. Respiratory specimens are notoriously difficult for the lab to interpret. The results are extremely specimen dependent. Colonization of the airways by different bacteria that don't necessarily cause infection but that grow on the plates can cloud the issue of what the true pathogen is.
Get her an infectious disease doc and/or pulmonologist ASAP.