Longer naps don't come in until much later - 4 to 6 months ish. The one long lunchtime nap from about 12 months.
At 5 weeks your DD is still in the newborn phase of if not feeding, then sleeping so should be either already asleep or trying to get to sleep most of the time.
That will start to reduce after 6 weeks, but I still wouldn't expect more than 90 minutes between sleeps.
Shorter regular naps through the day are more usual, say between 20 minutes to 90 minutes in every 2 hours.
Good sleep promotes better sleep and unfortunately poor sleep promotes worse sleep. It sounds like your DD is not getting sleep regularly enough through the day. Babies at this age cannot 'save up' their tiredness to have a bigger sleep, then need naps.
In terms of structure (I'm a massive fan of structure), you can still do this while being baby lead with the EASY method. Be lead by the baby in terms of how long a cycle lasts, but throughout the day follow:
Eat (have a feed)
Activity (have a short time awake)
Sleep (have a sleep)
You (this stands for you having some time to yourself - also makes the acronym EASY)
My DD is 4 months and followed the EASY method in 90 minute cycles from birth. The length of the cycle has increased in time (lead by my daughter, not forced by me) and she now follows in 2 1/2 hour cycles. But it makes the day structured and predictable. Also ensures regular naps and feeds.