You can visit the Masjid Negara, or National Mosque of Malaysia during certain times of the day between prayer service (best to look the visitation times up before going; for my trip the non-Muslim visiting hours were Monday to Sunday: 9am-12pm, 3pm-4pm, 5:30pm-6:30pm, except Friday which is 3pm-4pm, 5:30pm-6:30pm only). It is a beautiful example of modernist Islamic architecture. I found it to be a wonderful place of refuge, calm and quiet with devoted practitioners and subdued non-Muslim visitors wearing purple robes (which one can borrow for free to ensure your legs are covered and for women, hair, chest and arms are covered).
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It is located at the Kuala Lumpur train station and next to the KTM headquarters, which are also interesting works of Islamic archecture in themselves. From the train station, take the underground walkway located on the north-west end of the station (turn left when you leave the main exit) to cross the busy street between the station at the Masjid. It’s also easy walking distance from the Hotel Majestic, where I was staying.
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Inside the Masjid Negara:
The main prayer hall: 
Up the hill is the stunning Islamic Art Museum, which was a highlight of my visit to KL. The museum itself is a work of art:
The dome…
And the ceiling inside: 
With a wonderfully diverse mix of Islamic art from all over the world in various forms, such as this ceramic plate from Central Asia: 
And detail of a prayer mat from Turkey: 
The work displayed was intricate, requiring great skill and beautiful materials, and quietly elegant. It ranged from calligraphy in microscopic pocket books and delicate jewelry to carved wooden doors and huge metal swords and chain-mail.
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While the museum exhibits could use more cohesive organization and flow, I learned so much about the diversity of Islamic communities, especially in Central and SE Asia, which I knew very little about before, through descriptions of their clothing, furniture, prayer and ceremonial objects.