Yeah, just keep to the 5k mark at the moment for next 4-8 weeks and strengthen first, then slowly add distance at 5% per week or 10% in 2 weeks.
Say for your Z2 runs, that means 5.5k 2 weeks later
6km 4 weeks later
6.6k 6 weeks later
7.26k 8 weeks later
8k 10 weeks later
8.8k 12 weeks later
9.68k 14 weeks later
10.64 16 weeks later
11.71k 18 weeks later
12.88k 20 weeks later.
14.16k 22 weeks later
15.6k 24 weeks later.
17.14k 26 weeks later
18.85k 28 weeks later
20.74k 30 weeks later.
Distances agaration can aleady, no need to be too exact, just for illustration....
That means total duration is conservatively 34-38 weeks with first 4-8 weeks focusing on solely strengthening. You do your zone 2 runs and also any form of speedwork along the way keeping number of Z2 running / speedwork days the same.
The above schedule also drills and reinforces consistency/persistence/discipline for mindset, which is key in SG's context with work and career. If coupled with interest, these alone would ensure very good success in performance, not merely "finishing the half marathon" or any other distance for that matter. Seriously, even my pri 6 school kid can technically finish HM lah, it's actually no biggie if no time limit....he actually hit 96km cycling distance yesterday and on Saturday 68km on a cheap foldie, no road bike or what lol But then he would struggle in maintaining say 30km/h even for just 2 mins ie along the 1.5k length at Bay Garden East. As long as sufficient fuel, coz he has bonked several times before, once so bad that he had to sit down beside the road and then slowly walk to the bus stop and lie down while i searched for soda drinks. lol.
You should largely be fine even with somewhat bad form doing self-coached way with no analysis by other experts , things will adapt ok + help from strenghthening along the way.
edit - a bit of correction regarding "no" time limit. I really meant no time goals.
Usually locally is 4hrs cut off for HM which is very very generous and designed to be extremely inclusive to even full distance walkers, which is like just a bit faster than 11:30 min/km pace.
That is something that a pri school kid or BMI 25-30 pax can walk at...less any fainiting spells or heat adaptation issues. Of coz if one is already injured or morbidly obese BMI 35 or a fella who is aged 60 and have had zero exercise for the past 4 decades, then cannot lah.