More power will require a bigger core.
Look at their other reference designs for guidance.
IF you could use a higher frequency you could get more epower from the same core BUT the IC used is fixed at 72 kHz.
iW1677 offline PWM controller data sheet here.
Other datasheets from same manufacturer here
The other ICs with similar functionality also seem to use similar or lower frequencies (sample only checked).
Core volume is about inverse linear with frequency, can change drastically with topology - if energy is stored in the core then core size and power handling are linear (flyback, boost, ...) while if the core is used in the "forward mode (forward converter, push pull, ...) with the core storing magnetisation energy but not storing all the power being transferred per cycle then the energy per core size can be greater.
In flyback designs, which the examples below both are, the energy is stored in the core on the input half cycle and delivered to the output on the next half cycle. (I say half cycle in each case but each part cycle will probably not be of equal duration). The core is driven until the ampere turns of magnetisation (amps x number of primary turns) is such that the core just starts to "saturate". This is the limit beyond which little or no extra energy can be stored in the core.
You can assume with good likelihood of being correct that the designer has used the core to near it's full ampere-turn capacity, so that you will need a larger core to get more power.
The easiest solution is to look at the other reference designs from the same manufacturer.
Their EBC10007 reference design is rated at 2.1A at 5v, or slightly more than double your chosen example .
Your 003 design uses Core: EEM12.4 (ferrite material TDK PC40 or equivalent)
The 007 design uses Core: EPC17 (ferrite material TDK PC40 or equivalent)
The photos in the design documents make the difference in core size clear and you will be able to search for core dimensions. Topologies are essentially the same in both circuits - look at the coil dottings on input and output coils - when Vin has +ve applied to the primary dot the undotted secondary connection is applied to the output diode so it is not conducting SO the energy must be stored in the transformer SO this is a flyback converter in both cases.
Note that the 5W design uses a bipolar transistor and the 10W design uses a MOSFET driver (Q1 in both cases) and there are other differences, but the general guidelines provided by these examples will be about right.
You should do your own research to confirm the TDK core sizes, but the following appears very liable to be useful:
TDK EEM12.x from here.
The number after EEM appears to be the major core dimension - see data sheet.
TDK EPC17 from here